Kirchhoff's Junction Rule (Current Law):

Sum of currents entering junction equals sum leaving:

$\sum I_{in} = \sum I_{out}$

Or: $\sum I = 0$

Conservation of charge: charge cannot accumulate at junction.

Kirchhoff's Loop Rule (Voltage Law):

Sum of voltages around any closed loop equals zero:

$\sum V = 0$

Physical meaning: electric field is conservative (work around closed loop = 0)

Sign convention:

• Voltage rise: positive (going from - to + across battery)
• Voltage drop: negative (going across resistor in direction of current)

represent magnetic field direction and strength.

Properties:

• Form closed loops (no beginning/end)
• Never cross
• Closer together = stronger field
• Outside magnet: from N to S pole
• Inside magnet: from S to N pole

Patterns:

• Bar magnet: dipole pattern
• Current loop: circular around wire
• Solenoid: uniform inside, return outside

Long straight wire:

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

Where:

• $\mu_0 = 4\pi \times 10^{-7}$ T - m/A (permeability of free space)
• I = current
• r = distance from wire

Current loop (center):

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

Where R = loop radius

Solenoid (inside):

$B = \mu_0 n I = \mu_0 \frac{N}{L} I$

Where:

• n = turns per unit length
• N = total turns
• L = length

Direction of magnetic field around current-carrying wire.

Procedure:

1. Point thumb in direction of current
2. Fingers curl in direction of B-field

For solenoid:

• Fingers curl in direction of current in coils
• Thumb points to N pole (direction of B inside)

Magnetic force on moving charge:

$\vec{F} = q\vec{v} \times \vec{B}$

Magnitude: $F = qvB\sin\theta$

Direction: given by right-hand rule

• Point fingers in direction of v
• Curl toward B
• Thumb points to F (for positive charge)

Properties:

• Force perpendicular to both v and B
• Does NO work (W = 0, since F ⟂ v)
• Only changes direction, not speed
• For negative charge, force opposite direction

Force on current-carrying conductor in magnetic field:

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

Magnitude: $F = BIL\sin\theta$

Direction: right-hand rule

• Fingers in direction of current
• Curl toward B
• Thumb points to F

Applications:

• Electric motors (force on current loop)
• Loudspeakers
• Railguns

Torque on current loop (motor principle): $\tau = NIAB\sin\theta$

Where N = number of turns, A = loop area

For force on moving charge or current in B-field:

1. Point fingers in direction of velocity/current
2. Curl fingers toward magnetic field direction
3. Thumb points in direction of force (for positive charge)

For negative charges (electrons):

• Force is opposite to thumb direction

Alternative for current-carrying wire:

• Palm faces B-field
• Fingers point in current direction
• Thumb points in force direction

Perpendicular to v and B (circular motion):

Centripetal force provided by magnetic force: $qvB = \frac{mv^2}{r}$

Radius of circular path: $r = \frac{mv}{qB}$

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

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

Parallel to B: no force, straight line motion At angle to B: helical motion (circular + linear)

through surface:

$\Phi = \vec{B} \cdot \vec{A} = BA\cos\theta$

Where:

• B = magnetic field
• A = area
• $\theta$ = angle between B and normal to area

Units: Weber (Wb) = T - m2

Physical meaning:

• Measure of magnetic field passing through area
• Maximum flux when B parallel to area normal
• Zero flux when B perpendicular to area

Changing flux causes induced emf.

Changing magnetic flux induces emf:

$\mathcal{E} = -\frac{d\Phi}{dt}$

For N turns: $\mathcal{E} = -N\frac{d\Phi}{dt}$

Ways to change flux:

1. Change B (vary field strength)
2. Change A (move coil, expand/contract)
3. Change $\theta$ (rotate coil)

Magnitude of induced emf proportional to rate of flux change.

Direction of induced current opposes change that produced it.

$\mathcal{E} = -\frac{d\Phi}{dt}$

(negative sign is Lenz's law)

Applications:

• Induced B-field opposes original change
• Energy conservation: induced current creates magnetic field opposing change

Procedure:

1. Determine direction of flux change (increasing/decreasing)
2. Induced field must oppose this change
3. Right-hand rule to find current direction

EMF induced by conductor moving through magnetic field:

$\mathcal{E} = BLv$

Where:

• B = magnetic field
• L = length of conductor
• v = velocity perpendicular to B

Derivation from Faraday's law: $\mathcal{E} = -\frac{d\Phi}{dt} = -\frac{d(BA)}{dt} = -B\frac{dA}{dt} = -B(L dx/dt) = -BLv$

Direction given by right-hand rule (Lenz's law ensures opposition)

Applications:

• Generators
• Railguns
• Velocity sensors