AP Physics B & C - Magnetic Field - Equations to be Remembered

The section ‘Magnetic Fields’ in the AP Physics syllabus contains the following sub sections:

(1) Forces on moving charges in magnetic fields

(2) Forces on current-carrying wires in magnetic fields

(3) Fields of long current-carrying wires

(4) Biot-Savart’s law and Ampere's law (For AP Physics C only)

AP Physics B carries 4% of the total points in this section while AP Physics C carries 10%.

Here are the equations to be remembered in this section:

(1) The magnetic force ‘F’ acting on a particle of charge ‘q’ moving with velocity ‘v’ making an angle ‘θ’ with a magnetic field ‘B’ (fig.) is given by

F = qvB sinθ

The force F is perpendicular to both v and B. In vector form, the above equation is

F = qv×B

Note that bold face characters are used to represent vectors.

When electric and magnetic fields act simultaneously on a charge, the total force on the charge is given by Lorentz force equation,

F = q(v×B + E) where E is the electric field.

(2) The path of a charged particle of mass ‘m’ projected with a velocity ‘v’ perpendicular to a magnetic field B is a circle of radius ‘r’ given by

qvB = mv²/r, where ‘q’ is the charge

[Note that we have equated the magnetic force to the centripetal force]

Therefore, r = mv/qB

If the particle is projected into the magnetic field at an angle other than zero or 90º, the path is a helix of radius ‘r’ given by

r = mv sinθ/qB

(3) The period of circular motion as well as helical motion of a charged particle in a magnetic field is

T = 2πm/qB

(4) The frequency of revolution along the circular path (or helical path) is

f = qB/2πm

This is called the cyclotron frequency

(5) The magnetic force ‘dF’ acting on an elemental length dℓ of a conductor carrying a current ‘I’ placed in a magnetic field B, making an angle ‘θ’ with the magnetic field is given

dF = IdℓB sinθ

The force dF is perpendicular to both dℓ and B. In vector form, the above equation is

dF = I dℓ×B, treating the elemental length dℓ as a vector.

In the case of a straight conductor of length ℓ, the magnetic force is

F= IℓB sinθ which can be written in vector form as

F = I ℓ×B

(6) Force per unit length between two infinitely long parallel current carrying conductors is given by

F = µ₀I₁I₂/2πd, where µ₀ is the permeability of free space, ‘d’ is the separation between the conductors and I₁ and I₂ are the currents in the conductors.

(7) Torque on a plane current carrying coil (current loop) placed in a magnetic field B is

τ = nIAB sinθ, where ‘n’ is the number of turns in the coil, A is the area of the coil, I is the current in the coil and θ is the angle between the area vector and the magnetic field vector. Remember that the area is a vector which has magnitude equal to the area and direction perpendicular to the area.

(8) Magnetic field (dB) due to a current element of length dℓ at a point P distant ‘r’ from the current element is given by Biot-Savart law:

dB = (µ₀/4π) Idℓ sinθ/r²

This magnetic field is perpendicular to the plane containing the current element and the point P.

In vector notation the above equation is

dB = (µ₀/4π) Idℓ ×r/r³

Here r is a vector of length r directed from the current element to the

point P.

The length of the current element also is treated as a vector dℓ of length dℓ with its direction same as that of the current.

(9) The magnetic field due to a straight infinitely long current carrying conductor
at a point P at a perpendicular distance ‘r’ from the conductor is

B =µ₀I/2πr

(10) The magnetic field due to a plane circular current carrying

coil of ‘n’ turns and radius R at a point P on the axis at a distance

‘x’ from the centre of the coil is

B = µ₀nR²I /2(R² + x²)^3/2

The magnetic field at the centre of the coil is

B= µ₀nI/2R

(11) The magnetic field on the axis of an infinitely long straight solenoid at a point P well within the solenoid is

B= µ₀nI, where ‘n’ is the number of turns per metre of the solenoid.

(12) The magnetic field inside a toroid of ‘n’ turns per metre is

B= µ₀nI

(13) Ampere’s circuital law states that the line integral of magnetic flux density over any closed curve is equal to µ₀ times the total current passing through the surface enclosed by the closed curve. This is stated mathematically as

∫B. dℓ = µ₀I (The integration is over the closed path)

Amperes circuital law as modified by Maxwell to accommodate the displacement current flowing through even free space is

∫B. dℓ = µ₀ [I+ ε₀ (dφ_E/dt)], where ε₀ (dφ_E/dt) is the displacement current resulting from the rate of change of electric flux φ_E. ε₀ is the permittivity of free space.

In the next post we will discuss questions in this section.