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Moving Charges and Magnetism

Physics • Chapter 4
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Oersted's Experiment

Discovered the link between electricity and magnetism. A current-carrying wire deflects a magnetic compass needle.

Oersted's Experiment

Conclusion: Moving charges (current) produce a magnetic field in the surrounding space.

Lorentz Force

Magnetic Force

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

Direction given by Fleming's Left Hand Rule or Right Hand Cross Product.

Total Lorentz Force

Force in presence of both E and B fields:

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

Motion in B-Field

  • v || B ($\theta=0$): Linear path.
  • v $\perp$ B ($\theta=90^\circ$): Circular path. $r = \frac{mv}{qB}$
  • Arbitrary angle: Helical path.
Particle Trajectories

Force on a Conductor

A wire carrying current $I$ in a magnetic field experiences a force:

$$ \vec{F} = I(\vec{l} \times \vec{B}) $$ $$ F = IlB \sin\theta $$
Force on Conductor

Biot-Savart Law

Magnetic field due to a small current element $Id\vec{l}$:

$$ d\vec{B} = \frac{\mu_0}{4\pi} \frac{I(d\vec{l} \times \hat{r})}{r^2} $$
Biot Savart Law

Field on Loop Axis

Magnetic field on the axis of a circular loop of radius $R$ at distance $x$:

$$ B = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}} $$
Field on Axis

At center ($x=0$): $B = \frac{\mu_0 I}{2R}$

Ampere's Circuital Law

Line integral of magnetic field around a closed loop equals $\mu_0$ times total current enclosed.

$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed} $$
Ampere Law

Field in a Solenoid

Inside a long straight solenoid:

$$ B = \mu_0 n I $$

where $n$ = number of turns per unit length.

Solenoid Field

Force: Parallel Wires

Force per unit length between two parallel wires separated by $d$:

$$ \frac{F}{l} = \frac{\mu_0 I_1 I_2}{2\pi d} $$
Parallel Wires Force
  • Same direction: Attract
  • Opposite direction: Repel

Torque on a Loop

A current loop in a uniform magnetic field experiences a torque:

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

$\vec{m} = NIA$ (Magnetic Dipole Moment)

Torque on Loop

Moving Coil Galvanometer

Principle

Current loop in radial magnetic field experiences deflecting torque: $NIAB$. Restoring torque: $k\phi$.

$$ \phi = \left(\frac{NAB}{k}\right) I $$
Galvanometer