Moving Charges and Magnetism

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Moving Charges and Magnetism — Key Facts

Magnetic Force on a Moving Charge: $$\vec{F} = q(\vec{v} \times \vec{B})$$

Force magnitude: $F = qvB\sin\theta$

Direction: given by Fleming’s left-hand rule (for positive charge; reverses for negative).

Lorentz Force: $$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$$

This combines electric and magnetic forces on a charge.

Cyclotron Motion: When v is perpendicular to B, charge moves in a circle: $$r = \frac{mv}{qB}$$ $$\omega = \frac{qB}{m}$$ $$T = \frac{2\pi m}{qB}$$ (time period, independent of speed!)

⚡ JEE Exam Tip: Magnetic force does NO work because F is always perpendicular to v. It only changes direction, not speed. This is why cosmic rays spiral rather than spiral inward.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Biot-Savart Law:

Magnetic field due to current element Idl at distance r: $$d\vec{B} = \frac{\mu_0}{4\pi}\frac{Id\vec{l} \times \hat{r}}{r^2}$$

Direction: given by right-hand grip rule (thumb = current direction, fingers curl in direction of B).

Biot-Savart for Common Shapes:

Shape	Field
Straight wire (distance d)	$B = \frac{\mu_0 I}{2\pi d}$
At centre of circular loop (radius R)	$B = \frac{\mu_0 I}{2R}$
Solenoid (inside, n turns/unit length)	$B = \mu_0 n I$
At axis of circular loop (distance x)	$B = \frac{\mu_0 IR^2}{2(R^2 + x^2)^{3/2}}$

Force on Current-Carrying Conductor: $$\vec{F} = I(\vec{L} \times \vec{B})$$

For straight wire in uniform B: $F = BIL\sin\theta$

Force between Parallel Currents: $$F/L = \frac{\mu_0}{2\pi}\frac{I_1 I_2}{d}$$

• Parallel currents in same direction: attract
• Parallel currents in opposite directions: repel

⚡ JEE Exam Tip: 1 ampere is defined such that when two parallel wires carrying 1 A each are 1 metre apart, the force per metre is $2 \times 10^{-7}$ N/m. This definition also gives $\mu_0 = 4\pi \times 10^{-7}$ H/m.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Torque on Current Loop:

For rectangular loop with N turns in uniform magnetic field: $$\tau = NIAB\sin\theta$$

where θ = angle between plane of loop and magnetic field

Magnetic dipole moment: $\vec{m} = NIA\hat{n}$ (direction given by right-hand rule)

This is analogous to electric dipole p = qd.

Moving Coil Galvanometer:

Used to detect small currents. Working: coil experiences torque, spring provides restoring torque.

$$\theta = \frac{NAB}{k} \cdot I$$

where k = torsional constant of spring

Conversion to Ammeter: connect low resistance shunt $R_s = \frac{I_g R_g}{I - I_g}$ Conversion to Voltmeter: connect high resistance $R = \frac{V}{I_g} - R_g$

Helical Motion:

When velocity has component parallel to B, charge moves in a helix:

• Parallel component: $v_\parallel = v\cos\alpha$
• Perpendicular component: $v_\perp = v\sin\alpha$ (causes circular motion)
• Pitch (distance along field per turn): $p = v_\parallel T = \frac{2\pi m v_\parallel}{qB}$
• Radius of helix: $r = \frac{m v_\perp}{qB}$

Magnetic Field of Spinning Charged Sphere:

For a uniformly charged sphere rotating with angular velocity ω: $$B_{axis} = \frac{\mu_0 \omega R \sigma}{2} = \frac{\mu_0 q \omega}{2\pi R}$$

This models planetary magnetic fields and spinning charged bodies.

Vector Potential:

For uniform B field: $\vec{A} = \frac{1}{2}(\vec{B} \times \vec{r})$ (about any origin)

In quantum mechanics: momentum operator $\hat{p} = -i\hbar\vec{\nabla} \rightarrow$ replaced by $(\vec{p} - q\vec{A})$ in magnetic field.

Cyclotron Frequency (Advanced):

The cyclotron frequency is independent of speed (non-relativistically). This is used in:

• Particle accelerators
• Mass spectrometers
• NMR (nuclear magnetic resonance)

Relativistic correction: as speed approaches c, frequency decreases. synchrotron adjusts frequency to keep particle in sync.

⚡ JEE Advanced 2022 Analysis: Questions on helical motion, torque on current loop, and conversion of galvanometer appeared in 2022. For magnetic moment calculations, remember m = NIA for a planar loop, and the direction is perpendicular to the plane (follow right-hand rule: fingers in direction of current, thumb gives m direction).

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