Force on moving charge in electric and magnetic fields:

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

For magnetic field only: $\vec{F} = q\vec{v} \times \vec{B}$

Magnitude: $F = qvB\sin\theta$

Where:

• q = charge
• $\vec{v}$ = velocity
• $\vec{B}$ = magnetic field
• $\theta$ = angle between v and B

Direction: given by right-hand rule (for positive charge)

Perpendicular to field (v ⊥ B):

• Circular motion (magnetic force provides centripetal)
• Speed constant (force ⟂ velocity -> no work)
• Path: circle

Parallel to field (v ∥ B):

• No force (sinθ = 0)
• Straight line motion

At angle to field:

• Helical motion (circular + linear)

When v ⟂ B:

Magnetic force provides centripetal force: $qvB = \frac{mv^2}{r}$

Cyclotron radius: $r = \frac{mv}{qB}$

Period of rotation: $T = \frac{2\pi r}{v} = \frac{2\pi m r}{mvr} = \frac{2\pi m}{qB}$

Period independent of speed (important property).

Cyclotron frequency: $f = \frac{1}{T} = \frac{qB}{2\pi m}$

Angular frequency: $\omega = 2\pi f = \frac{qB}{m}$

Device that selects particles with specific velocity.

Use crossed E and B fields where electric and magnetic forces balance:

$F_E = F_B$ $qE = qvB$

$v = \frac{E}{B}$

Only particles with this velocity pass through undeflected.

Procedure:

1. Particles enter region with perpendicular E and B
2. Forces oppose each other
3. Only particles with v = E/B go straight
4. Others are deflected

Device for separating ions by mass.

Principle:

1. Accelerate ions through potential V: $qV = \frac{1}{2}mv^2$
2. Velocity: $v = \sqrt{2qV/m}$
3. Ions enter magnetic field, follow circular path: $r = \frac{mv}{qB}$

Combining: $r = \frac{m}{qB}\sqrt{\frac{2qV}{m}} = \frac{\sqrt{2mV}}{qB}$

Radius depends on mass: heavier ions have larger radii

Applications:

• Isotope separation
• Chemical analysis
• Identifying unknown masses

At distance r from long straight wire carrying current I:

$B = \frac{\mu_0 I}{2\pi r}$

Where:

• $\mu_0$ = 4π × 10-7 T - m/A (permeability of free space)
• I = current (A)
• r = perpendicular distance from wire (m)

Direction: right-hand grip rule

• Thumb in direction of current
• Fingers curl in direction of B-field

Field strength ∝ 1/r (like electric field from line charge)

Circular loop center (N turns, radius R):

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

Direction: perpendicular to loop plane (right-hand rule)

On axis (distance x from center): $B = \frac{\mu_0 N I R^2}{2(R^2 + x^2)^{3/2}}$

At center (x = 0): reduces to above formula.

Inside ideal solenoid:

$B = \mu_0 n I$

Where:

• n = number of turns per unit length (turns/m)
• I = current (A)
• $\mu_0$ = 4π × 10-7 T - m/A

Properties:

• Uniform field inside
• Parallel to axis
• Direction: right-hand grip rule

Outside solenoid: B ≈ 0 (field lines return through outside)

Finite solenoid: near ends, field reduced (fringing effect)

Force on current-carrying conductor in magnetic field:

$\vec{F} = I\vec{L} \times \vec{B}$

Magnitude: $F = BIL\sin\theta$

Where:

• I = current (A)
• L = length of wire in field (m)
• B = magnetic field (T)
• $\theta$ = angle between current direction and B

Direction: right-hand rule

• Fingers in direction of current
• Curl toward magnetic field
• Thumb points in force direction

Magnetic dipole moment: $\vec{\mu} = NIA\hat{n}$

Where:

• N = number of turns
• I = current
• A = loop area
• $\hat{n}$ = unit normal to loop

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

Magnitude: $\tau = NIAB\sin\theta$

Where $\theta$ = angle between dipole moment and magnetic field

Maximum torque when loop plane ∥ B-field Zero torque when $\vec{\mu} \parallel \vec{B}$ (equilibrium)

$U = -\vec{\mu} \cdot \vec{B} = -\mu B\cos\theta$

Minimum energy when $\dagger \parallel \vec{B}$ (stable equilibrium) Maximum energy when opposite to B (unstable equilibrium)

Magnetic field from current element:

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

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

Where:

• I = current
• $d\vec{l}$ = current element vector
• $\hat{r}$ = unit vector from element to point
• r = distance from element

Direction: perpendicular to both current element and position vector

For finite straight wire carrying current I:

At perpendicular distance a from wire, field at point opposite middle:

$B = \frac{\mu_0 I}{4\pi a}(\sin\theta_1 + \sin\theta_2)$

Where angles from wire ends to point.

Special cases:

• Infinite wire: $\sin\theta_1 = \sin\theta_2 = 1$ -> $B = \frac{\mu_0 I}{2\pi a}$
• Semi-infinite wire: one angle = 1, other < 1

For circular loop (radius R, current I) at distance x from center:

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

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

For N turns: $B = \frac{\mu_0 N I}{2R}$

Line integral of magnetic field around closed loop equals mu0 times enclosed current.

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$

Where:

• $\oint$ = closed loop integral
• $I_{enc}$ = total current passing through loop
• $\mu_0$ = 4π × 10-7 T - m/A

Physical meaning: circulation of B depends only on enclosed current.

Choose Amperian loop to match symmetry.

Guidelines:

• Loop should be closed
• Choose loop where B is constant on each part or parallel to loop
• Match current distribution symmetry

Symmetry types:

• Circular symmetry about wire: circular Amperian loop
• Solenoidal symmetry: rectangular loop inside solenoid

Using Ampere's law with circular Amperian loop radius r:

$\oint \vec{B} \cdot d\vec{l} = B \cdot 2\pi r = \mu_0 I_{enc}$

$B = \frac{\mu_0 I_{enc}}{2\pi r}$

For uniform current distribution in wire of radius R:

• Outside (r > R): $I_{enc} = I$
• Inside (r < R): $I_{enc} = I(r/R)^2$ (proportional to enclosed area)

Using rectangular Amperian loop parallel to axis:

Inside: $B = \mu_0 n I$

Where n = turns per unit length.

Outside: B ≈ 0 (for ideal infinite solenoid)

Derivation: B constant and parallel to axis inside; field outside negligible.

For toroid (doughnut shape) with N turns, radius r from center:

$B = \frac{\mu_0 N I}{2\pi r}$

Valid inside toroid; outside B = 0.

Toroidal symmetry: circular field lines inside toroid.