Comprehensive Guide to Electromagnetic Induction and Magnetism and Inductive Circuits

Torque and Magnetic Dipole Moments

• Motors represent the most frequent application of magnetic force acting upon current-carrying wires. When current passes through loops, the magnetic field exerts torque, which causes the rotation of a shaft. This transformation process converts electrical energy into mechanical work.

• The torque $\tau$ acting on a coil consisting of $N$ loops, where each loop has an area $A$ and carries a current $I$, in an external uniform magnetic field $B$ is determined by the equation: $\tau = NIAB \sin(\theta)$ In this expression, $\theta$ represents the angle between the normal to the plane of the loop and the magnetic field direction.

• In vector notation, torque is expressed as: $\mathbf{\tau} = N I \mathbf{A} \times \mathbf{B}$ Here, $\mathbf{A}$ is a vector oriented perpendicular to the loop's plane with a magnitude equivalent to the area of the loop.

• The magnetic dipole moment of a coil is defined as the quantity $IA$.

• The potential energy $U$ associated with a magnetic field is calculated using: $U = -\mathbf{\mu} \cdot \mathbf{B}$

Galvanometers and Sensitivity Factors

• A galvanometer contains wire loops wound on a soft iron core. This core pivots between the pole faces of a permanent magnet. A current passing through the coil generates torque, while a small spring provides a restoring torque.

• Current sensitivity in a meter is defined as the deflection per unit current: $\text{Sensitivity} = \frac{\phi}{I} = \frac{NAB}{k}$

• To enhance the sensitivity of a current meter, the following adjustments can be made:

  • Increase the magnetic field strength $B$.

  • Decrease the spring constant $k$ (using weaker springs).

  • Increase the number of turns $N$ in the coil.

  • Increase the area of the coil $A$.

• Voltmeter sensitivity is defined as the deflection per unit potential difference: $\text{Sensitivity (Voltmeter)} = \frac{\phi}{V}$ Given the relationship $V = IR$, this can also be expressed as: $\frac{\phi}{V} = \frac{NAB}{kR}$

Ammeters, Voltmeters, and Resistance Conversion

• An ammeter is designed to measure the electric current flowing through it, whereas a voltmeter measures the potential difference between two specific points.

• A milliammeter can be converted into an ammeter capable of measuring higher currents by connecting a low-resistance resistor, known as a shunt ($R_{sh}$), in parallel. If $I$ is the total current, $I_c$ is the current through the moving-coil ammeter, and $I_{sh}$ is the current through the shunt: $I = I_{sh} + I_c$ The relationship for the shunt is: $I_c R_c = I_{sh} R_{sh}$

• To convert a milliammeter into a voltmeter, a high-resistance resistor, known as a multiplier ($R_s$), is connected in series with the milliammeter. If $V$ is the total potential difference and $R_c$ is the resistance of the milliammeter: $V = I_c (R_s + R_c)$

Ampere's Law and the Biot-Savart Law

• Ampere's Law states that the line integral of the magnetic field vector $\mathbf{B}$ around any closed path is equal to $\mu_0$ (permeability of free space) multiplied by the total current $I$ flowing through the circuit: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I$

• The Biot-Savart Law, formulated by Jean-Baptiste Biot (1774-1862) and Felix Savart (1791-1841), describes the magnetic field produced by current elements. The field $d\mathbf{B}$ at a point $P$ due to a current element $I d\mathbf{l}$ is: $d\mathbf{B} = \frac{\mu_0}{4 \pi} \frac{I d\mathbf{l} \times \hat{\mathbf{r}}}{r^2}$

• If $\theta$ is the angle between the current element and the line joining it to point $P$, the magnitude is: $dB = \frac{\mu_0}{4 \pi} \frac{I dl \sin(\theta)}{r^2}$

Magnetic Field Configurations

• Long Straight Wire: At a perpendicular distance $a$ from a wire carrying current $I$: $B = \frac{\mu_0 I}{2 \pi a}$

• Center of a Circular Loop: For a loop of radius $r$ carrying current $I$: $B = \frac{\mu_0 I}{2 r}$

• Axis of a Circular Loop: At a distance $x$ from the center of a loop with radius $a$: $B = \frac{\mu_0 I a^2}{2 (a^2 + x^2)^{3/2}}$ At the center ($x = 0$), this returns to $B = \frac{\mu_0 I}{2 a}$.

• Solenoid: At the center of a solenoid with $N$ turns and length $L$, carrying current $I$: $B = \mu_0 n I$ Where $n = N/L$ is the linear turn density. At either end of the solenoid, the field is: $B_{end} = \frac{1}{2} \mu_0 n I$

• Right-Hand Source Rule: To determine the direction of $B$, grasp the wire with the right hand so the thumb points in the direction of the current $I$; the curled fingers indicate the sense of the magnetic field.

Forces Between Parallel Conductors

• When two parallel wires carry currents $I_1$ and $I_2$ at a distance $d$ apart, the field from one exerts a force on the other. The force per unit length is defined as: $\frac{F_B}{\ell} = \frac{\mu_0 I_1 I_2}{2 \pi d}$

• Interaction rules:

  • Conductors carrying currents in the same direction attract each other.

  • Conductors carrying currents in opposite directions repel each other.

• Definition of the Ampere: When the force per unit length between two long, parallel wires separated by $1\,m$ carrying identical currents is $2 \times 10^{-7}\,N/m$, the current in each wire is defined as $1\,A$.

Faraday’s Law and Lenz’s Law

• Faraday's Law of Induction: A changing magnetic field induces an electromotive force (EMF). The induced EMF in a wire loop is proportional to the rate of change of magnetic flux through the loop: $\epsilon = -N \frac{\Delta \Phi_B}{\Delta t}$

• Magnetic Flux: Defined as the total number of field lines passing through a loop: $\Phi_B = BA \cos(\theta)$ Unit: Weber ($Wb$). $1\,Wb = 1\,T \cdot m^2$.

• Lenz's Law: The minus sign in Faraday's law indicates that the induced current generates a magnetic field that tends to oppose the change in the original magnetic field.

• Flux changes can occur through:

  • Changing the magnetic field strength $B$.

  • Changing the area of the loop $A$.

  • Changing the angle $\theta$ between the loop and the field.

Motional EMF and Electric Fields

• EMF in a Moving Conductor: For a conductor of length $l$ moving with velocity $v$ perpendicular to a magnetic field $B$: $\epsilon = Blv$ The induced current tends to slow the moving bar, requiring an external force to maintain motion. This principle is utilized in the measurement of blood velocity.

• Electric Field Generalization: A changing magnetic flux induces an electric field regardless of whether conductors are present.

Electric Generators

• A generator transforms mechanical energy into electrical energy using an axle rotated by external forces like steam or falling water.

• AC Generator: Uses slip rings to maintain contact. The induced EMF is: $\epsilon = NBA\omega \sin(\omega t) = \epsilon_0 \sin(\omega t)$ Where $\epsilon_0 = NBA\omega$ is the peak EMF.

• DC Generator: Uses a split-ring commutator. The commutator reverses the current every half revolution to ensure the torque maintains rotation in the same direction.

Back EMF and Eddy Currents in Motors

• In a motor, rotation within a magnetic field induces an EMF called back EMF ($\epsilon$), which opposes the supply voltage $V$. The armature current is: $I = \frac{V - \epsilon}{R_a}$ Where $R_a$ is the armature resistance.

• Mechanical power developed in the motor is calculated as: $P_{mech} = \epsilon I$

• In a generator, the induced current produces a counter torque, requiring increased external torque to maintain rotation.

Transformers and Power Transmission

• A transformer consists of primary and secondary coils linked by an iron core. Transformers require AC to function.

• EMF Ratio: $\frac{E_s}{E_p} = \frac{N_s}{N_p}$

• Current Ratio (assuming no losses): $\frac{I_p}{I_s} = \frac{N_s}{N_p}$

• Types:

  • Step-up: N_s > N_p, resulting in E_s > E_p.

  • Step-down: N_s < N_p, resulting in E_s < E_p.

Inductance: Self and Mutual

• Self Inductance ($L$): A varying current in a coil induces an EMF in itself: $\epsilon = -L \frac{dI}{dt}$ Unit: Henry ($H$). $1\,H = 1\,V \cdot s/A = 1\,\Omega \cdot s$.

• Mutual Inductance ($M$): A changing current in one coil induces an EMF in a secondary coil: $\epsilon_2 = -M \frac{dI_1}{dt}$ and $\epsilon_1 = -M \frac{dI_2}{dt}$

Energy Stored in Magnetic Fields

• Energy can be stored in the magnetic field of an inductor: $U = \frac{1}{2} L I^2$

• For a solenoid, energy density (energy per unit volume) is: $u = \frac{B^2}{2 \mu_0}$

LR Circuits: Growth and Decay

• Time Constant (\tau): Defined as: $\tau = \frac{L}{R}$

• Current Growth: Upon closing a switch, current increases according to: $i = \frac{V}{R} (1 - e^{-t/\tau})$ At $t = \tau$, the current reaches approximately 63% of its final value ($I_0 = V/R$).

• Current Decay: If the battery is removed, current decreases following: $i = I_0 e^{-t/\tau}$ At $t = \tau$, the current decreases to about 37% of its original value.

Worked Examples

• Example 15.28 (Mechanical Power):

  • Armature resistance $R_a = 0.08\,\Omega$; Supply $V = 120\,V$; Current $I = 50\,A$.

  • Power supplied: $V \times I = 120 \times 50 = 6000\,W$.

  • Heat loss in resistance: $I^2 R_a = 50^2 \times 0.08 = 200\,W$.

  • Mechanical power = $6000 - 200 = 5800\,W$.

• Example 15.29 (Transformer Turns):

  • Step-down from $2.2\,kV$ to $110\,V$. Secondary turns $N_s = 25$.

  • $\frac{E_p}{E_s} = \frac{N_p}{N_s} \rightarrow \frac{2200}{110} = \frac{N_p}{25}$.

  • Primary turns $N_p = 500$.

• Example 15.49 (LR Circuit Current):

  • $R = 20\,\Omega$, $L = 0.30\,H$, $V = 90\,V$. Find current at $t = 0.05\,s$.

  • Time constant $\tau = L/R = 0.30/20 = 0.015\,s$.

  • $i = \frac{V}{R} (1 - e^{-t/\tau}) = \frac{90}{20} (1 - e^{-0.05/0.015})$.

  • $i = 4.5 (1 - e^{-3.33}) = 4.5 (1 - 0.0357) \approx 4.34\,A$.