Magnetic Fields and Forces

Magnetic flux density

Magnetic flux density (B) measures the strength and direction of a magnetic field, defined as the amount of magnetic flux (ϕ) passing through a unit area (A) perpendicular to the field.

B = ϕ/A

B = magnetic flux density (measure in T, Tesla)

where ϕ = magnetic flux (measured in webers, Wb)

and A = area (measured in m²)

Magnetic flux patterns due to a long straight wire

B = μ₀I / 2πr

where μ₀ = permeability of free space /Hm^-1

I = current/A

r = perperdicular distance from conductor/m

Magnetic field produced by a fl at circular coil

B = μ₀NI / 2r

Where N = number of turns

μ₀ = permeability of free space /Hm^-1

I = current/A

r = perperdicular distance from conductor/m

Magnetic field produced by a long solenoid

B = μ₀nI

where n = number of turns per unit length N/L

μ₀ = permeability of free space /Hm^-1

I = current/A

r = perperdicular distance from conductor/m

Force acting on a charge moving in a magnetic field

F = BQv

F – force on charge/N

B – magnetic flux density/T

Q – charge/C

v – speed/ms^−1

A particle with charge Q, travelling with a speed v at an angle of θ to the magnetic field

F = BQv sinθ

F – force on charge/N

B – magnetic flux density/T

Q – charge/C

v – speed/ms−1

θ – angle between v and B

Hall effect

A potential difference is set up transversely across a current-carrying conductor when a perpendicular magnetic field is applied.

What is a Hall Probe and how does it work?

• Purpose: Measures magnetic flux density (B).

• How it works:

  • Uses semiconductor slice (charge carriers have high drift velocity → larger Hall voltage).

  • Place semiconductor perpendicular to magnetic field.

  • Pass small constant current I through it.

  • Measure Hall voltage V_H.

• Formula:

  B=V_Hnet / I

  (Manufacturer gives n e t value; device is calibrated in known fields.)

What is a current balance and how does it work?

• Purpose: Measures magnetic flux density (B).

• How it works:

  • Wire frame (side AB placed perpendicular to field).

  • Current through frame creates downward force F on side AB.

  • Adjust rider mass mg until frame balances horizontally (pointer at zero).

  • Use principle of moments to find F.

Formula (from moments):

F=mg⋅y / x​

Then

B=F / Il

then

F = mgy / xlI

where l = length of AB, I = current, x and y are distances from pivot.

The work done on an electron by the electric field is

w = qV or w = eV (same thing)

Magnetic flux

a measure of the total magnetic field (B) passing through a given surface area (A)

Φ = Magnetic flux measured in webers (Wb)

B = magnetic flux density (Tesla/ T)

A = surface area (m²)

EMF (electromotive force) formula

E = V + Ir or E = I(R + r)

Potential divider

V_o = V_in (R₂/ R₁ + R₂)

Derive equation for resistance in series

In a series circuit, the same current I flows through all resistors. The total voltage across the combination equals the sum of the individual voltages (by Kirchhoff's 2nd Law):

V_total = V₁ + V₂ + V₃

Substituting V = IR for each:

IR_eff = IR₁ + IR₂ + IR₃

Dividing every term by the common current I:

R_eff = R₁ + R₂ + R₃

Derive equation for resistance in parallel

In a parallel circuit, all resistors share the same voltage V across them. The total current splits between branches (by Kirchhoff's 1st Law):

I_total = I₁ + I₂ + I₃

Substituting I = V/R for each branch:

V/R_eff = V/R₁ + V/R₂ + V/R₃

Dividing every term by the common voltage V:

1/R_eff = 1/R₁ + 1/R₂ + 1/R₃

Derive equation for capacitance in series

In a series circuit, charge is conserved through the process of electron flow, so the charge stored on each capacitor is the same: Q₁ = Q₂ = Q₃ = Q.

By Kirchhoff's 2nd Law, the total voltage equals the sum of individual voltages:

V = V₁ + V₂ + V₃

Since V = Q/C for each capacitor:

Q/C_total = Q/C₁ + Q/C₂ + Q/C₃

Dividing through by the common charge Q:

1/C_total = 1/C₁ + 1/C₂ + 1/C₃

Derive equation for capacitance in parallel

In a parallel circuit, all capacitors share the same potential difference V across them. The charges stored are different on each:

Q₁ = C₁V,   Q₂ = C₂V,   Q₃ = C₃V

The total charge is the sum of all individual charges:

Q_total = Q₁ + Q₂ + Q₃

Substituting:

Q_total = C₁V + C₂V + C₃V = V(C₁ + C₂ + C₃)

Since C = Q/V, dividing both sides by V:

C_total = C₁ + C₂ + C₃

Equation for force on a current-carrying conductor

F = BILsinθ

Faraday's Law

the magnitude of the induced e.m.f. is proportional to the rate of change of flux linkage.

ε = −N ΔΦ/Δt

ε = induced e.m.f. (V) ΔΦ/Δt = rate of change of flux (Wb·s⁻¹)

Lenz's Law

the direction of the induced e.m.f. (and hence induced current) is such that it opposes the change in flux that produced it

(it’s a consequence of conservation of energy. Explains the minus sign in Faraday's Law.)

Equation for EMF in a straigh conductor moving in a magnetic filed

ε = BLv   (straight conductor moving in field)

L = length of conductor (m) v = speed of conductor (m·s⁻¹)

• If the conductor does not move perpendicularly to the field, the formula is ε = BLvsinθ