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CLASS 12 • PHYSICS
Moving Charges & Magnetism
1. Field Laws
Biot-Savart Law
$dB = \frac{\mu_0 I dl \sin\theta}{4\pi r^2}$
Field at Center of Loop
$B = \frac{\mu_0 I}{2R}$
For N turns: Multiply by N
Field on Axis of Loop
$B = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$
2. Ampere's Law
Circuital Law
$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}$
Straight Wire
$B = \frac{\mu_0 I}{2\pi r}$
Solenoid (Inside)
$B = \mu_0 n I$
$n$ = turns per unit length ($N/L$)
Toroid
$B = \frac{\mu_0 N I}{2\pi r}$
3. Force on Charge
Lorentz Force
$\vec{F} = q(\vec{v} \times \vec{B}) = qvB\sin\theta$
Max Force at 90°, Zero at 0°
Circular Path (Cyclotron)
$r = \frac{mv}{qB}$
Time Period
$T = \frac{2\pi m}{qB}$
Independent of Speed (v) and Radius (r)
Kinetic Energy
$K = \frac{q^2 B^2 r^2}{2m}$
4. Force on Wire
Current Carrying Conductor
$\vec{F} = I(\vec{l} \times \vec{B}) = IlB\sin\theta$
Force Between Parallel Wires
$\frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d}$
Nature of Force
Same Direction Current: ATTRACT
Opposite Direction: REPEL
5. Torque on Loop
Magnetic Moment
$\vec{M} = NIA$
Torque
$\vec{\tau} = \vec{M} \times \vec{B} = MB \sin\theta$
$\theta$ is angle between Normal & B
Radial Field (Galvanometer)
$\tau = NIAB$
Here $\sin\theta = 1$ always
6. Galvanometer
Current Sensitivity
$I_s = \frac{\phi}{I} = \frac{NBA}{k}$
Voltage Sensitivity
$V_s = \frac{\phi}{V} = \frac{NBA}{kR}$
Conversion to Ammeter
$S = \frac{I_g G}{I - I_g}$
Shunt (S) in Parallel
Conversion to Voltmeter
$R = \frac{V}{I_g} - G$
Resistance (R) in Series
7. Common Exam Traps
- ⚠️ The Angle Trap: In $\tau = MB \sin\theta$, $\theta$ is angle with the AREA VECTOR (Normal), not the plane of the coil. If plane angle is given as $\alpha$, then $\theta = 90 - \alpha$.
- ⚠️ Parallel Wires: Unlike charges, 'Like' currents ATTRACT and 'Unlike' currents REPEL. Do not mix this up with Electrostatics!
- ⚠️ Work Done: Work done by Magnetic Force on a moving charge is ALWAYS ZERO (Force $\perp$ Velocity).
8. Golden Theory Rules
- Velocity Selector: If $v = E/B$, particle passes undeviated (Net Force = 0).
- Solenoid Ends: Field at the end of a long solenoid is exactly half of the center ($B = \mu_0 n I / 2$).
- Cyclotron Limit: Cannot accelerate electrons (mass too small) or neutrons (no charge).
