AP Physics C: E&M — Unit 4: Statics and Dynamics of Magnetic Fields

4.1 Forces on Moving Charges in Magnetic Fields

Defines of Magnetic Field ($\vec{B}$)

Magnetic Fields are vector fields produced by moving electric charges or intrinsic magnetic moments of elementary particles. Unlike electric fields which originate from static charges, magnetic fields have no "monopoles"; magnetic field lines always form closed loops.

• Symbol: $\vec{B}$
• SI Unit: Tesla (T). Note: $1 \text{ T} = 1 \frac{\text{N}}{\text{C} \cdot \text{m/s}} = 10^4 \text{ Gauss}$.
• Visual Representation: Field lines emerge from the North pole and enter the South pole. The density of lines indicates field strength.

The Lorentz Force

A charged particle moving through a magnetic field experiences a magnetic force. This is the fundamental interaction in magnetostatics.

$\vec{F}_B = q(\vec{v} \times \vec{B})$

Where:

• $q$ is the electric charge Magnitude and sign matter.
• $\vec{v}$ is the velocity vector of the particle.
• $\vec{B}$ is the magnetic field vector.
• $\times$ denotes the vector cross product.

Key Consequences Calculation:

1. Magnitude: $F_B = |q|vB \sin(\theta)$, where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$.
2. Zero Force Conditions: The force is zero if the particle is stationary ($v=0$) or moving parallel/anti-parallel to the field ($\sin(0^\circ) = 0$).
3. Direction: Perpendicular to the plane defined by $\vec{v}$ and $\vec{B}$. Determined by the Right-Hand Rule.
4. Work: The magnetic force is always perpendicular to motion ($ \vec{F}_B \perp \vec{v} $). Therefore, magnetic forces do NO work on charged particles ($ W = \int \vec{F} \cdot d\vec{r} = 0 $). They can change direction, but not speed or kinetic energy.

The Right-Hand Rule (RHR) for Forces

Used to determine the cross-product direction for a positive charge.

1. Point fingers of right hand in direction of velocity $\vec{v}$.
2. Curl fingers toward magnetic field $\vec{B}$.
3. Thumb points in direction of Force $\vec{F}_B$.

Note: For a negative charge (like an electron), find the direction for a positive charge and flip it 180 degrees.

Motion in a Uniform Magnetic Field

When a charged particle enters a uniform magnetic field with velocity perpendicular to the field, it undergoes Uniform Circular Motion.

• Centripetal Force: The magnetic force acts as the centripetal force.
  $F<em>B = F</em>c \implies qvB = \frac{mv^2}{r}$
• Radius of Path (Cyclotron Radius):
  $r = \frac{mv}{qB}$
• Period ($T$) & Frequency ($f$):
  $T = \frac{2\pi r}{v} = \frac{2\pi m}{qB}$
  $\omega = \frac{qB}{m} \quad (\text{Cyclotron Frequency})$

Helical Motion: If the velocity has a component parallel to the field ($v_{\parallel}$), the particle moves in a helix. The parallel component remains constant (pitch), while the perpendicular component drives the circular motion.

Applications

• Velocity Selector: Crossed $\vec{E}$ and $\vec{B}$ fields create opposing forces. Only particles with $v = \frac{E}{B}$ pass through undeflected ($qE = qvB$).
• Mass Spectrometer: Uses a velocity selector followed by a region of pure $\vec{B}$ field to separate ions by mass-to-charge ratio ($m/q$) based on the radius of curvature.

4.2 Forces on Current-Carrying Wires

Current is simply a collection of moving charges. Therefore, a wire carrying current in a magnetic field experiences a cumulative force.

Force Formula

For a straight wire of length $L$ carrying current $I$ in a uniform field:
$\vec{F}_B = I(\vec{L} \times \vec{B})$

For a curved wire or non-uniform field, we integrate along the path:
$\vec{F}_B = \int I (d\vec{l} \times \vec{B})$

• $\vec{L}$ or $d\vec{l}$ points in the direction of conventional current.

Maxwell's Note: If a wire forms a complete closed loop in a uniform magnetic field, the net force is zero.

Torque on a Current Loop

While the net force on a closed loop in a uniform field is zero, the net torque ($\tau$) is not. This is the principle behind electric motors.

$\vec{\tau} = \vec{\mu} \times \vec{B}$

• Magnetic Dipole Moment ($\vec{\mu}$): A vector defined for a current loop. $\vec{\mu} = N I A \, \hat{n}$
  • $N$: Number of turns.
  • $I$: Current.
  • $A$: Area of the loop.
  • $\hat{n}$: Normal vector (direction determined by RHR—curl fingers with current, thumb points to $\hat{n}$).

Potential Energy of a Dipole:
$U = -\vec{\mu} \cdot \vec{B} = -\mu B \cos(\theta)$

• Lowest energy (stable equilibrium) when $\vec{\mu}$ is parallel to $\vec{B}$.
• Highest energy (unstable) when anti-parallel.

4.3 Biot-Savart Law: Sources of Magnetic Fields

Just as Coulomb’s Law calculates the E-field from a point charge, the Biot-Savart Law calculates the B-field produced by a moving charge or current element.

General Formula

$d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{l} \times \hat{r}}{r^2}$

• $\mu_0$: Permeability of free space ($4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$).
• $d\vec{l}$: Infinitesimal length vector pointing in current direction.
• $\hat{r}$: Unit vector pointing from the wire element to the point of interest.
• $r$: Distance from the wire element to the point.

Important Derivations (AP C Learning Objectives)

1. Long Straight Wire

Integrating Biot-Savart for an infinite line of current yields:
$B = \frac{\mu_0 I}{2\pi r}$

• Direction: Circular loops around the wire (RHR: thumb along current, fingers curl along B).

2. Center of a Circular Loop (Radius $R$)

$B = \frac{\mu_0 I}{2R}$

3. On the Axis of a Current Loop

At a distance $z$ from the center of a loop of radius $R$:
$B<em>z = \frac{\mu</em>0 I R^2}{2(z^2 + R^2)^{3/2}}$

• At $z=0$, this reduces to the "center of loop" formula.
• At $z \gg R$, it approximates the field of a dipole ($B \propto 1/z^3$).

Force Between Parallel Wires

Two parallel wires exert magnetic forces on each other. Wire 1 creates a field at the location of Wire 2, which exerts a force on Wire 2.

$\frac{F}{L} = \frac{\mu<em>0 I</em>1 I_2}{2\pi d}$

• Parallel Currents: Attract.
• Anti-Parallel Currents: Repel.

4.4 Ampère’s Law

Ampère’s Law is the magnetic equivalent of Gauss’s Law. It is useful for finding magnetic fields in highly symmetric situations.

The Integral Form

$\oint \vec{B} \cdot d\vec{l} = \mu<em>0 I</em>{enc}$

• $\oint$: Line integral around a closed loop (Amperian loop).
• $d\vec{l}$: Vector along the path of the loop.
• $I_{enc}$: Net current penetrating the area defined by the loop.

Sign Convention: Curl your right fingers in the direction of integration ($d\vec{l}$). Your thumb defines the positive direction for current.

Applications of Ampère's Law

1. Long Straight Wire (Outside vs. Inside)

• Outside ($r > R$): We draw a circular loop radius $r$. $\oint B dl = B(2\pi r) = \mu0 I$. $B = \frac{\mu</em>0 I}{2\pi r}$
• Inside ($r < R$): Assuming uniform current density $J = I / (\pi R^2)$.
  $I<em>{enc} = J(\pi r^2) = I \left(\frac{r^2}{R^2}\right)$$B(2\pi r) = \mu</em>0 I \frac{r^2}{R^2} \implies B = \left( \frac{\mu_0 I}{2\pi R^2} \right) r$
  The field increases linearly from the center to the surface.

2. Solenoid

An ideal solenoid (long, tightly wound) has a uniform field inside and zero field outside.
$B = \mu_0 n I$

• $n = N/L$: Number of turns per unit length.
• The field is parallel to the axis. Direction given by RHR (curl fingers with current, thumb points North).

3. Toroid

A solenoid bent into a donut shape. Determine the field at a radius $r$ inside the coils ($N$ total turns).
$\oint \vec{B} \cdot d\vec{l} = B(2\pi r) = \mu<em>0 (NI)$$B = \frac{\mu</em>0 N I}{2\pi r}$

• The field is non-uniform (stronger at the inner radius, weaker at the outer radius).
• $B = 0$ outside the toroid entirely.

4.5 Common Mistakes & Pitfalls

1. Magnetic Work: A common trick question asks for the work done by the magnetic force on a charge moving in a circle. The answer is always Zero. Magnetic force changes direction, not Kinetic Energy.
2. Cross Product Order: $\vec{v} \times \vec{B} \neq \vec{B} \times \vec{v}$. The order matters. Reversing them flips the sign (direction). Always use $\vec{v}$ (fingers) then $\vec{B}$ (curl).
3. Electric vs. Magnetic Force: Electric force ($FE = qE$) acts on moving and stationary charges parallel to the field. Magnetic force ($FB = qv \times B$) acts only on moving charges perpendicular to the field.
4. $\,\mu0$ vs $\epsilon0$: Don't confuse the constants. $\mu0$ is for magnetism ($4\pi \times 10^{-7}$); $\epsilon0$ is for electricity.
5. Using Ampère's Law for Finite Wires: Ampère's law in the simple form $B(2\pi r) = \mu_0 I$ requires symmetry. You cannot use it easily for a short wire segment; you must use Biot-Savart for that.
6. Current Direction in Parallel Wires: Students often confusingly apply the "likes repel" rule from electrostatics to currents. For wires: Like currents Attract; Opposite currents Repel.