Electromagnetism: Magnetic Fields, Forces, and Induction

Fundamentals of Magnetism and Magnetic Fields

• Fundamental Law of Magnetism:

  • There exist two types of magnetic poles: North ($\text{N}$) and South ($\text{S}$).

  • Like poles repel one another ($\text{N}$ repels $\text{N}$, $\text{S}$ repels $\text{S}$).

  • Opposite poles attract one another ($\text{N}$ attracts $\text{S}$, $\text{S}$ attracts $\text{N}$).

  • Magnetic monopoles do not exist in isolation; cutting a magnet in half produces two smaller dipoles, each with its own North and South pole.

• Magnetic Field Vector ($\vec{B}$):

  • A magnetic field is a vector field that permeates the space around permanent magnets, electric currents, and moving charges.

  • The direction of the magnetic field vector ($\vec{B}$) at any point is defined as the direction in which the North ($\text{N}$) pole of an ideal test compass needle points when placed at that location.

  • Outside a magnet, magnetic field lines emerge from the North pole and curve around to enter the South pole ($\text{N} \rightarrow \text{S}$).

  • Inside a magnet, the field lines continue through the interior from the South pole back to the North pole ($\text{S} \rightarrow \text{N}$), forming continuous, closed loops.

  • Field line density corresponds directly to field intensity: regions where lines are closest together (such as at the pole faces) possess the greatest magnetic field strength.

• Earth's Magnetic Field:

  • Earth generates a dipolar magnetic field similar to that of a giant internal bar magnet tilted relative to the rotational axis.

  • The magnetic field is produced by convective electric currents flowing within Earth's molten iron-nickel outer core (the geodynamo mechanism).

  • Because the North pole of a navigational compass needle is attracted toward Earth's northern polar region, Earth's geographic North contains a magnetic South pole:

  • $\text{Geographic North} \approx \text{Magnetic South}$

  • $\text{Geographic South} \approx \text{Magnetic North}$

  • Field Orientation Relative to Earth's Surface:

  • Near the equator, the magnetic field lines run parallel to Earth's surface.

  • At the magnetic poles, the field lines are directed perpendicular to the surface (pointing vertically into the surface near the geographic North pole and pointing vertically outward near the geographic South pole).

Domain Theory and Ferromagnetism

• Domain Theory of Magnetism:

  • Materials that exhibit strong magnetic interactions and can become magnetized are classified as ferromagnetic materials.

  • Ferromagnetism originates at the atomic level from unpaired electron spins that produce microscopic magnetic dipole moments.

  • In ferromagnetic substances, quantum mechanical exchange interactions cause the magnetic moments of billions of adjacent atoms to align parallel to one another in localized regions termed magnetic domains (typically containing $10^{17}$ to $10^{21}$ atoms).

• States of Magnetic Alignment:

  • Non-ferromagnetic materials: Atoms do not possess net aligned magnetic moments; no magnetic domains exist.

  • Unmagnetized ferromagnetic materials: Magnetic domains exist, but their dipole orientations are randomly distributed throughout the crystal lattice. The vector sum of the internal magnetic fields cancels to zero, producing no net external field.

  • Magnetized ferromagnetic materials: When subjected to an external magnetic field ($\vec{B}_{\text{ext}}$), domains aligned with the field grow through domain wall migration, and other domains rotate to align parallel to $\vec{B}_{\text{ext}}$. The substance develops a macroscopic net magnetic field and becomes magnetized.

• Ferromagnetic Elements and Classifications:

  • The four elements that are ferromagnetic at room temperature are:

  • Iron ($\text{Fe}$)

  • Nickel ($\text{Ni}$)

  • Cobalt ($\text{Co}$)

  • Gadolinium ($\text{Gd}$)

  • Numerous specialized alloys (e.g., Alnico, neodymium-iron-boron, permalloy) also exhibit strong ferromagnetism.

  • Soft Ferromagnetic Materials (e.g., pure annealed iron):

  • Exhibit high magnetic permeability but low retentivity.

  • Readily gain domain alignment in the presence of an external field and rapidly lose their magnetization when the field is removed.

  • Ideal for temporary magnets, transformer cores, and electromagnet cores.

  • Hard Ferromagnetic Materials (e.g., carbon steel, rare-earth alloys):

  • Require strong external fields to align domains (high coercivity), but retain their alignment indefinitely after the external field is removed.

  • Ideal for manufacturing permanent bar magnets, compass needles, and electric motor stators.

• Demagnetization and Environmental Effects:

  • Thermal Agitation: Heating a permanent magnet increases atomic kinetic vibrations. Above the substance's characteristic Curie temperature, thermal motion completely disrupts domain alignment, returning the material to an unmagnetized (paramagnetic) state.

  • Mechanical Shock: Dropping or striking a permanent magnet imparts mechanical energy that shakes the domains loose from their locked orientations, allowing them to randomize and cancel out.

  • Magnetization of Structural Beams: Steel construction beams frequently develop a measurable permanent magnetic field. Mechanical vibrations caused by riveting, welding, and pneumatic drilling agitate domains while the beams reside within Earth's ambient magnetic field, facilitating domain alignment with Earth's field lines.

• Theoretical and Quantitative Applications:

  • Deflection of an Iron Ball by a Magnet:

  • An iron ball of mass $m = 72\,\text{g} = 0.072\,\text{kg}$ is suspended from a string of length $l = 15\,\text{cm}$ and deflected by $\theta = 20^\circ$ from the vertical toward a South pole positioned at equal height.

  • Domain Explanation: The external field from the South pole aligns the iron ball's domains such that the side nearest the magnet becomes an induced North pole, creating an attractive magnetic force ($F_m$).

  • Equilibrium Force Balance:       $\Sigma F_y = T \cos(\theta) - F_g = 0 \implies T = \frac{mg}{\cos(\theta)}$       $\Sigma F_x = T \sin(\theta) - F_m = 0 \implies F_m = mg \tan(\theta)$       $F_m = (0.072\,\text{kg})(9.81\,\text{m/s}^2) \tan(20^\circ) = 0.257\,\text{N} \approx 0.26\,\text{N}$

  • Distance Dependence of a Dipolar Field:

  • The on-axis magnetic field strength of a bar magnet at a distance $D$ from its center is modeled by:       $B = \frac{k}{4\pi D^3}$

  • To obtain a linear graph from experimental field data, plot $B$ on the vertical axis versus $\frac{1}{D^3}$ on the horizontal axis.

  • The slope of the resulting line corresponds directly to the physical constant:       $\text{Slope} = \frac{k}{4\pi}$

Electromagnetism and Hand Rules for Currents and Solenoids

• Oersted's Discovery (1820):

  • Hans Christian Oersted demonstrated that electric charges in motion produce a surrounding magnetic field.

  • Stationary charges produce only an electrostatic field ($\vec{E}$), whereas moving charges produce both an electric field ($\vec{E}$) and a magnetic field ($\vec{B}$).

• Three-Dimensional Vector Notation:

  • Out of the page (toward observer): Represented by a dot or circle with a dot ($\odot$ or $\bullet$), depicting the point of an approaching arrow.

  • Into the page (away from observer): Represented by a cross or circle with an X ($\otimes$ or $\times$), depicting the tail feathers of a receding arrow.

• Charge Conventions for Hand Rules:

  • Left-Hand Rules (LHR): Must be used when analyzing electron flow ($e^-$) or the motion of any negative charge.

  • Right-Hand Rules (RHR): Must be used when analyzing conventional current ($I$) or the motion of positive charges ($p^+$, alpha particles).

• Hand Rule #1: Magnetic Field Surrounding a Straight Conductor:

  • Thumb: Points in the direction of charge motion / electric current.

  • Curled Fingers: Wrap around the conductor pointing in the direction of the concentric, circular magnetic field lines ($\vec{B}$).

• Hand Rule #2: Solenoids and Electromagnets:

  • A solenoid is a cylindrical coil of wire consisting of numerous closely spaced turns.

  • Curled Fingers: Wrap around the solenoid coils in the direction of current flow through the individual wire segments.

  • Thumb: Points in the direction of the internal magnetic field ($\vec{B}_{\text{inside}}$), which points directly toward the North ($\text{N}$) pole of the electromagnet.

• Electromagnet Performance and Core Insertion:

  • An air-core solenoid generates a uniform axial magnetic field in its interior.

  • Inserting a soft ferromagnetic core (such as soft iron) into the core dramatically increases the magnetic field strength.

  • The solenoid's magnetic field aligns the domains within the iron bar; the magnetic field of the aligned domains adds vectorially to the solenoid's field, multiplying total flux density.

• Solenoid Field Equation and Graphical Modeling:

  • The magnetic field magnitude inside an ideal long solenoid is given by:     $B = \frac{\mu_0 N I}{L}$

  • $B$ = internal magnetic field strength ($\text{T}$)

  • $\mu_0 = 1.26 \times 10^{-7}\,\text{T}\cdot\text{m/A}$ (magnetic permeability of free space)

  • $N$ = total number of wire turns/loops

  • $I$ = current flowing through the coil ($\text{A}$)

  • $L$ = length of the solenoid ($\text{m}$)

  • When graphing $B$ versus $I$ for a solenoid of length $L = 11.0\,\text{cm} = 0.110\,\text{m}$:

  • Measured data:

    • $I = 0.80\,\text{A} \rightarrow B = 1.65 \times 10^{-4}\,\text{T}$

    • $I = 1.60\,\text{A} \rightarrow B = 3.25 \times 10^{-4}\,\text{T}$

    • $I = 2.40\,\text{A} \rightarrow B = 5.85 \times 10^{-4}\,\text{T}$

    • $I = 3.20\,\text{A} \rightarrow B = 5.60 \times 10^{-4}\,\text{T}$

    • $I = 4.00\,\text{A} \rightarrow B = 7.55 \times 10^{-4}\,\text{T}$

    • $I = 4.80\,\text{A} \rightarrow B = 8.50 \times 10^{-4}\,\text{T}$

  • Best-fit linear slope: $\text{Slope} = 1.72 \times 10^{-4}\,\text{T/A}$

  • Theoretical slope relation: $\text{Slope} = \frac{\mu_0 N}{L}$

  • Number of turns ($N$):       $N = \frac{\text{Slope} \cdot L}{\mu_0} = \frac{(1.72 \times 10^{-4}\,\text{T/A})(0.110\,\text{m})}{1.26 \times 10^{-7}\,\text{T}\cdot\text{m/A}} = 150\,\text{loops}$

• Comparative Field Properties:

  • Gravitational Field ($\vec{g}$):

  • Source: Mass.

  • Direction: Universally attractive (directed radially inward toward the mass center).

  • Detection: Measured by observing gravitational force on an invariant test mass.

  • Electric Field ($\vec{E}$):

  • Source: Electric charge (stationary or moving).

  • Direction: Directed radially outward from positive charges and inward toward negative charges.

  • Detection: Measured by the electrostatic force vector acting on a small positive test charge ($q^+$).

  • Magnetic Field ($\vec{B}$):

  • Source: Permanent magnetic dipoles or moving electric charges / currents.

  • Direction: Continuous loops; points from North to South outside a dipole and South to North inside.

  • Detection: Measured by the deflection torque acting on the North pole of a test compass needle or the deflecting force on a moving charge.

Magnetic Force on Conductors and Current-Carrying Wires

• Newton's Third Law and Magnetic Interactions:

  • Because a current-carrying wire produces a magnetic field that exerts a magnetic force on an external permanent magnet, Newton's Third Law demands that the external magnet exert an equal in magnitude, opposite in direction magnetic force on the current-carrying wire.

• Hand Rule #3 for Conductors (Magnetic Force on Current):

  • Thumb: Points in the direction of electric current (use Left Hand for electron flow $e^-$, Right Hand for conventional current $I$).

  • Extended Straight Fingers: Point in the direction of the external magnetic field ($\vec{B}_{\text{ext}}$).

  • Palm: Faces outward in the direction of the resulting magnetic force vector ($\vec{F}_m$).

• Magnitude of the Conductor Magnetic Force:

  • The magnetic deflecting force on a straight conductor is given by:     $F_m = I l_\perp B = I l B \sin(\theta)$

  • $F_m$ = magnetic deflecting force ($\text{N}$)

  • $I$ = electric current flowing through the wire ($\text{A}$)

  • $l$ = physical length of the conductor residing directly within the magnetic field ($\text{m}$)

  • $B$ = magnitude of the external magnetic field ($\text{T}$)

  • $\theta$ = angle between the conductor axis and the magnetic field vector

  • Boundary Conditions:

  • When the wire runs completely perpendicular to the field ($\theta = 90^\circ$, $\vec{l} \perp \vec{B}$), $\sin(90^\circ) = 1$, yielding maximum magnetic force ($F_{m,\text{max}} = I l B$).

  • When the wire runs parallel or anti-parallel to the field ($\theta = 0^\circ$ or $180^\circ$, $\vec{l} \parallel \vec{B}$), $\sin(\theta) = 0$, yielding zero magnetic force ($F_m = 0$).

• Definition of Current and Charge Equivalence:

  • Electric current is the temporal rate of electric charge transfer through a conductor cross-section:     $I = \frac{q}{t}$

  • $q = n e$ (where $n$ is the total number of charge carriers and $e = 1.60 \times 10^{-19}\,\text{C}$ is the elementary charge magnitude)

  • $t$ = elapsed time interval ($\text{s}$)

• Equivalence of the Tesla ($\text{T}$):

  • Rearranging the force expression for $B$ establishes the SI unit equivalents:     $1\,\text{T} = 1\,\frac{\text{N}}{\text{A}\cdot\text{m}} = 1\,\frac{\text{N}}{\left(\frac{\text{C}}{\text{s}}\right)\cdot\text{m}} = 1\,\frac{\text{N}\cdot\text{s}}{\text{C}\cdot\text{m}}$

• Current Balance and Magnetic Levitation Problems:

  • Suspended Conductor Equilibrium:

  • A horizontal wire of mass $m = 55\,\text{g} = 0.055\,\text{kg}$ and active length $l = 8.0\,\text{cm} = 0.080\,\text{m}$ is suspended at rest inside a horizontal magnetic field.

  • Charge transfer: $n = 7.3 \times 10^{20}\,e^-$ travel through the wire in $t = 13\,\text{s}$.

  • Current magnitude:       $q = (7.3 \times 10^{20})(1.60 \times 10^{-19}\,\text{C}) = 116.8\,\text{C}$       $I = \frac{116.8\,\text{C}}{13\,\text{s}} = 8.985\,\text{A}$

  • Force balance for levitation ($F_m = F_g$):       $I l B = mg \implies B = \frac{mg}{I l} = \frac{(0.055\,\text{kg})(9.81\,\text{m/s}^2)}{(8.985\,\text{A})(0.080\,\text{m})} = 0.75\,\text{T}$

  • Current Balance Scale Dynamics:

  • A $10.0\,\text{cm}$ horizontal conductor is hung from an overhead nonconductive scale. With zero current, the balance reads $0.450\,\text{N}$ (the wire's true weight, $F_g$).

  • Wire mass: $m = \frac{0.450\,\text{N}}{9.81\,\text{m/s}^2} = 0.04587\,\text{kg} = 45.9\,\text{g}$.

  • Sending $I = 5.00\,\text{A}$ increases the reading to $0.750\,\text{N}$. The magnetic force acts downwards:       $F_m = F_{\text{scale}} - F_g = 0.750\,\text{N} - 0.450\,\text{N} = 0.300\,\text{N}$

  • Magnetic field strength:       $B = \frac{F_m}{I l} = \frac{0.300\,\text{N}}{(5.00\,\text{A})(0.100\,\text{m})} = 0.600\,\text{T}$

  • Reversing the magnetic field orientation directs $F_m$ upward, yielding a new scale reading:       $F_{\text{scale}} = F_g - F_m = 0.450\,\text{N} - 0.300\,\text{N} = 0.150\,\text{N}$

  • Conductor Accelerated Vertically:

  • A wire carrying $n$ electrons over time $t$ in a perpendicular field $B$ undergoes an upward acceleration $a$ against gravity:       $F_{\text{net}} = F_m - F_g \implies M a = I L B - M g$       $M(a + g) = \left(\frac{n e}{t}\right) L B \implies M = \frac{B n e L}{(a + g) t}$

• Interactions Between Parallel Current-Carrying Conductors:

  • Currents in the Same Direction (Parallel):

  • Applying Hand Rule #1 reveals that between the two wires, their respective magnetic fields point in opposite directions and partially cancel.

  • The higher external field density outside pushes the two wires together; like currents attract.

  • Currents in Opposite Directions (Anti-parallel):

  • Between the two wires, their respective magnetic fields point in the same direction, reinforcing one another and concentrating flux lines.

  • The repulsion between concentrated field lines forces the wires apart; opposite currents repel.

Magnetic Force on Freely-Moving Charged Particles

• Lorentz Magnetic Force on a Point Charge:

  • The magnetic deflecting force acting on a discrete electric charge moving through an external magnetic field is given by:     $F_m = q v_\perp B = q v B \sin(\theta)$

  • $F_m$ = magnetic force magnitude ($\text{N}$)

  • $q$ = charge magnitude ($\text{C}$)

  • $v$ = particle speed relative to the field ($\text{m/s}$)

  • $B$ = external magnetic field strength ($\text{T}$)

  • $\theta$ = angle between particle velocity vector ($\vec{v}$) and magnetic field vector ($\vec{B}$)

  • If $\vec{v} \parallel \vec{B}$ ($\theta = 0^\circ$ or $180^\circ$), $F_m = 0$; the particle continues in straight-line motion unaffected by the field.

  • If $\vec{v} \perp \vec{B}$ ($\theta = 90^\circ$), the particle experiences maximum magnetic deflecting force ($F_m = q v B$).

• Hand Rule #3 for Point Charges:

  • Thumb: Velocity vector ($\vec{v}$) of the moving charge.

  • Straight Fingers: External magnetic field vector ($\vec{B}_{\text{ext}}$).

  • Palm: Direction of the resulting magnetic force vector ($\vec{F}_m$).

  • Use the Left Hand for negative charges ($e^-$, anions) and the Right Hand for positive charges ($p^+$, alpha particles $\alpha^{2+}$, cations).

• Uniform Circular Motion in Perpendicular Fields:

  • Because the magnetic force vector is mutually perpendicular to both velocity and magnetic field ($\vec{F}_m \perp \vec{v}$ and $\vec{F}_m \perp \vec{B}$), the force does no work on the charged particle:     $W = \vec{F}_m \cdot \Delta \vec{d} = F_m \Delta d \cos(90^\circ) = 0$

  • Consequently, the magnetic force alters only the direction of the velocity vector while leaving orbital speed, kinetic energy ($E_k = \frac{1}{2}mv^2$), and momentum magnitude constant.

  • When entering a uniform perpendicular magnetic field, the constant-magnitude deflecting force acts strictly as a centripetal force ($F_m = F_c$):     $q v B = \frac{m v^2}{r}$

• Orbital Radius of Curvature Formulas:

  • Solving the centripetal equilibrium for radius ($r$):     $r = \frac{m v}{q B}$

  • When the charged particle is accelerated from rest across an electric potential difference ($V$) before entering the magnetic field:     $\Delta E_k = W_e \implies \frac{1}{2} m v^2 = q V \implies v = \sqrt{\frac{2 q V}{m}}$

  • Substituting speed into the radius expression:     $r = \frac{m}{q B} \sqrt{\frac{2 q V}{m}} = \frac{1}{B} \sqrt{\frac{2 m V}{q}}$

• Synthesized Problems and Particle Dynamics:

  • Heavily Ionized Particle Deflection:

  • A particle with $7$ excess electrons ($q = 7(1.60 \times 10^{-19}\,\text{C}) = 1.12 \times 10^{-18}\,\text{C}$) is accelerated from rest across voltage $V$ to $v = 4.00 \times 10^6\,\text{m/s}$ West, then enters a $B = 300\,\text{mT} = 0.300\,\text{T}$ field directed into the page.

  • Deflection trajectory diameter $d = 14.0\,\text{cm} \implies r = 0.0700\,\text{m}$.

  • Applying LHR #3: Thumb West, Fingers Into Page, Palm pushes North (deflecting clockwise).

  • Inertial Mass Calculation:       $m = \frac{q B r}{v} = \frac{(1.12 \times 10^{-18}\,\text{C})(0.300\,\text{T})(0.0700\,\text{m})}{4.00 \times 10^6\,\text{m/s}} = 5.88 \times 10^{-27}\,\text{kg}$

  • Accelerating Potential ($V$):       $V = \frac{m v^2}{2 q} = \frac{(5.88 \times 10^{-27}\,\text{kg})(4.00 \times 10^6\,\text{m/s})^2}{2(1.12 \times 10^{-18}\,\text{C})} = 4.20 \times 10^4\,\text{V} = 42.0\,\text{kV}$

  • Energy and Work Values:       $W_{\text{electric}} = \Delta E_k = q V = (1.12 \times 10^{-18}\,\text{C})(4.20 \times 10^4\,\text{V}) = 4.70 \times 10^{-14}\,\text{J}$       $W_{\text{magnetic}} = 0\,\text{J}$

  • Total Impulse During Acceleration:       $J = \Delta p = m \Delta v = (5.88 \times 10^{-27}\,\text{kg})(4.00 \times 10^6\,\text{m/s}) = 2.35 \times 10^{-20}\,\text{N}\cdot\text{s}\text{ West}$

  • Mass-Acceleration Inverse Proportionality:

  • For ions possessing identical charge and speed ($q, v$) entering a uniform field ($B$):       $F_m = q v B = m a \implies a = (q v B) \frac{1}{m} \implies a \propto \frac{1}{m}$

  • Plotting acceleration ($a$) versus reciprocal mass ($\frac{1}{m}$) yields a straight line with slope:       $\text{Slope} = q v B$

  • Using slope $6.77 \times 10^{-15}\,\text{N}$, with $v = 4.00 \times 10^6\,\text{m/s}$ and single electron deficit ($q = 1.60 \times 10^{-19}\,\text{C}$):       $B = \frac{\text{Slope}}{q v} = \frac{6.77 \times 10^{-15}\,\text{N}}{(1.60 \times 10^{-19}\,\text{C})(4.00 \times 10^6\,\text{m/s})} = 1.06 \times 10^{-2}\,\text{T}$

Electromagnetic Induction and Lenz's Law

• Faraday's Law of Induction:

  • An electric potential difference (electromotive force, $\text{EMF}$) and associated electric current are induced in a conducting circuit whenever the magnetic flux through the circuit changes over time.

  • Magnetic flux ($\Phi$) represents the net quantity of magnetic field lines traversing the area enclosed by a circuit loop ($\Phi = B_\perp A$).

  • Current can be induced by:

  • Altering the strength of the magnetic field ($B$).

  • Changing the surface area of the loop exposed to the field ($A$).

  • Changing the orientation angle between the magnetic field and the surface area.

• Lenz's Law:

  • The polarity of an induced electromotive force produces an induced current whose own magnetic field vector ($\vec{B}_{\text{ind}}$) creates a flux that directly opposes the change in original magnetic flux ($\Delta \Phi$) that created it.

  • Conservation of Energy Justification: Lenz's Law guarantees that mechanical energy must be expended to induce electrical power. The induced magnetic field always creates a mechanical retarding force ($\vec{F}_{m(\text{ind})}$) that resists the relative physical motion. This negative work transforms the mechanical energy of motion directly into electrical energy.

  • If the induced magnetic field acted to assist the flux change, the resulting force would accelerate the moving object spontaneously, producing an endless increase in kinetic and electrical energy without work input, violating the Law of Conservation of Energy.

• Method for Determining Induced Current Direction:

  • 1. Determine the direction of the external magnetic field vector ($\vec{B}_{\text{ext}}$).

  • 2. Determine whether magnetic flux through the loop is increasing (approaching magnet) or decreasing (receding magnet).

  • 3. Establish the required direction of the induced magnetic field ($\vec{B}_{\text{ind}}$) using Lenz's Law:

  • If flux is increasing: $\vec{B}_{\text{ind}}$ points in the opposite direction of $\vec{B}_{\text{ext}}$ to oppose the increase.

  • If flux is decreasing: $\vec{B}_{\text{ind}}$ points in the same direction as $\vec{B}_{\text{ext}}$ to oppose the decrease.

  • 4. Apply Hand Rule #2 to the coil: Point the thumb in the direction of $\vec{B}_{\text{ind}}$ (toward the induced North pole); the curled fingers indicate the direction of the induced electric current through the individual coil windings.

• Magnet Dropped Through a Hollow Conducting Copper Tube:

  • A cylindrical permanent magnet with its South pole oriented downward is dropped from rest through a vertical, non-magnetic copper tube.

• Eddy Current Analysis:

  • Below the falling magnet: Downward-directed South flux is increasing. To oppose this approach, the tube develops circular circulating currents (eddy currents) that produce a South magnetic pole directed upward toward the magnet's falling South pole. Like poles repel, exerting an upward retarding force.

  • Above the falling magnet: North flux is decreasing (receding). The tube generates eddy currents producing an attractive South pole at the top of the tube that pulls upward on the magnet's receding North pole.

  • Net Effect: Both regions produce an upward-directed magnetic force ($F_{m(\text{ind})}$) opposing the downward pull of gravity.

• Kinematic Evolution of Motion:

  • Initial Release ($t = 0$): Speed $v = 0$, so rate of flux change is zero, resulting in zero induced magnetic force ($F_m = 0$). The magnet is in true free fall ($a = g = 9.81\,\text{m/s}^2$ downward).

  • Intermediate Acceleration ($0 < v < v_t$): As velocity increases, the rate of flux change accelerates, increasing induced current and magnetic force ($F_m \propto v$). The net downward force shrinks ($F_{\text{net}} = F_g - F_m$), causing downward acceleration to decrease monotonically toward zero ($a < g$).

  • Terminal Velocity State ($v = v_t$): The upward induced magnetic force becomes exactly equal in magnitude to the downward gravitational weight ($F_m = F_g$). Net force is zero ($F_{\text{net}} = 0$), downward acceleration ceases ($a = 0$), and the magnet descends at a constant terminal velocity ($v_t$).

• Scale Reading Behavior: If the copper tube rests on a force scale, the reading begins at the tube's static weight ($F_{\text{scale}} = F_{g,\text{tube}}$). As the magnet enters and falls, the scale reading increases by Newton's Third Law ($F_{\text{scale}} = F_{g,\text{tube}} + F_m$) until terminal velocity is attained, at which point the scale measures the combined weight of the tube and the magnet ($F_{\text{scale}} = F_{g,\text{tube}} + F_{g,\text{magnet}}$).

• Methods to Reduce Terminal Velocity: Terminal velocity ($v_t$) can be reduced by:

  • Using a permanent magnet with greater magnetic field strength.

  • Increasing tube wall conductivity (e.g., cooling the copper or using ultra-pure copper).

  • Reducing the radial clearance gap (narrower tube diameter) between the magnet and the tube inner wall.

DC Electric Motors, Generators, and Motional EMF

• The Motor Effect:

  • When an electric current is passed through a conducting loop (armature) situated inside a magnetic field, the segments of the loop running perpendicular to the field experience opposing magnetic deflecting forces.

  • These opposing linear forces produce a net mechanical torque around the central axis, driving rotational motion.

• Fundamental Energy Conversions:

  • DC Motor: Transforms electrical energy into mechanical kinetic energy.

  • Generator: Transforms mechanical kinetic energy into electrical energy.

• Operating Cycle of a DC Electric Motor:

  • Essential Components: Stationary field magnets, rotating armature loop, split-ring commutator (C-rings), stationary carbon brushes, external DC voltage source.

• Step 1 (Peak Torque Position): The split-ring commutator segments (C-rings) maintain physical contact with the carbon brushes, completing the DC circuit. Current flows in opposite directions on the two lateral sides of the armature. Applying Hand Rule #3 reveals that one side experiences an upward magnetic force while the opposite side experiences a downward magnetic force. This force couple delivers maximum rotational torque, spinning the armature.

• Step 2 (Inertial Transition through Vertical Gap): When the armature rotates through the vertical plane perpendicular to the field lines, the split gaps of the commutator pass over the brushes. Electrical contact is broken temporarily; current halts ($I = 0$) and magnetic force vanishes ($F_m = 0$). Rotational inertia (angular momentum) carries the armature smoothly past this dead zone.

• Step 3 (Commutation and Continuous Unidirectional Torque): As the armature advances past the vertical position, each commutator half strikes the opposing brush. This reverses the physical direction of current flow through each segment of the armature coil. By reversing current at every half-rotation ($180^\circ$), the magnetic force direction on each segment reverses, ensuring that the net torque continues acting in the same rotational direction. The cycle repeats continuously.

  • Motional Electromotive Force ($\text{EMF}$):

• When a conductor of physical length $l$ moves through an external magnetic field $B$ at velocity $v$, the free charge carriers inside the conductor experience a Lorentz deflecting force along the wire length.

• This charge migration establishes a potential difference across the ends of the conductor known as motional $\text{EMF}$ ($V_{\text{ind}}$):     $V_{\text{ind}} = B_\perp l v = B l v \sin(\theta)$

• Maximum motional $\text{EMF}$ occurs when the magnetic field, conductor length, and velocity vector are all mutually orthogonal ($\vec{B} \perp \vec{l} \perp \vec{v}$).

• If any two vectors run parallel to each other ($\vec{v} \parallel \vec{B}$, $\vec{l} \parallel \vec{B}$, or $\vec{l} \parallel \vec{v}$), the induced potential difference drops to zero ($V_{\text{ind}} = 0$).

• Current in Closed Circuits (Ohm's Law):

  • Connecting the moving conductor to a closed circuit of total resistance $R$ establishes a closed-loop current:     $I_{\text{ind}} = \frac{V_{\text{ind}}}{R} = \frac{B_\perp l v}{R}$

  • Worked Example: A conducting rod of active length $l = 8.00\,\text{cm} = 0.0800\,\text{m}$ moves at $v = 22.0\,\text{m/s}$ through a perpendicular field $B = 0.700\,\text{T}$ across a circuit with resistance $R = 140\,\text{m}\Omega = 0.140\,\Omega$:     $V_{\text{ind}} = (0.700\,\text{T})(0.0800\,\text{m})(22.0\,\text{m/s}) = 1.232\,\text{V}$     $I_{\text{ind}} = \frac{1.232\,\text{V}}{0.140\,\Omega} = 8.80\,\text{A}$

• Loop Entering, Transiting, and Exiting a Magnetic Field:

  • Stage 1: Loop Entering the Field:

  • Magnetic flux is increasing inward ($\Delta \Phi > 0$).

  • By Lenz's Law, the loop generates an induced current to oppose this flux growth.

  • The induced current flowing across the leading edge experiences a magnetic braking force ($F_m = I l B$) directed opposite to velocity, causing the loop's speed to decline.

  • Stage 2: Loop Fully Enclosed Within the Uniform Field:

  • Because the field is spatially uniform, the total enclosed flux remains constant ($\Delta \Phi = 0$).

  • The motional $\text{EMF}$ generated across the leading edge is canceled by the identical motional $\text{EMF}$ generated across the trailing edge.

  • Net induced potential difference is zero; no current flows ($I = 0$).

  • With zero current, magnetic braking vanishes ($F_m = 0$), and the loop glides at constant velocity.

  • Stage 3: Loop Exiting the Field:

  • Magnetic flux is decreasing ($\Delta \Phi < 0$).

  • The loop establishes an induced current in the opposite direction to reinforce the decaying flux.

  • The trailing edge still in the field experiences an opposing magnetic force ($F_m$) directed against the velocity, decelerating the loop further until it exits the field completely.

• Free-Falling Rectangular Loop Induction:

  • A rectangular wire coil with width $x$, length $y$, and total resistance $R_T$ falls from rest under gravity through a height $h$ before its bottom edge enters a perpendicular field $B$.

  • Impact speed entering the field: $v = \sqrt{2 g h}$.

  • Induced motional electromotive force: $V_{\text{ind}} = B x v = B x \sqrt{2 g h}$.

  • Resulting induced electric current:     $I = \frac{V_{\text{ind}}}{R_T} = \frac{B x \sqrt{2 g h}}{R_T}$