A-level Physics/Forces, Fields and Energy/Electromagnetism

Magnetic Force on a Current

The formula below calculates the force being undergone by a current in a magnetic field.

$F=BIL\sin(\theta )$

F is the force being undergone by the current - measured in Newtons(N).

B is the magnetic field strength (or flux density) of the field the charge is in (I.e The number of magnetic field lines per unit area) - measured in Teslas (T).

I is current of the current - measured in amperes (A).

L is the length of conductor in the magnetic field - measured in meters (m)

θ is the angle the current makes with the magnetic field - measured in either radians or degrees (° or ^C)

The force undergone by the current is at its maximum magnitude when the current vector is perpendicular to the magnetic field vector (ie. Force is maximum when θ = 90°) as Sin(90°) = 1. However the force undergone by the current is at a magnitude of 0 N when the current vector is parallel to the magnetic field vector (ie. Force is 0 N when θ = 0°) as Sin(0°) = 0.

The direction of the Force may be found by application of the Right-hand rule rule as follows:

1. The index finger points in the direction of the velocity vector, v.
2. The middle finger points in the direction of the magnetic field vector B.
3. The thumb points in the direction of the cross product of v and B, which is equal to the force vector, F.

For example, for a positively charged particle moving to the north, in a region where the magnetic field points west, the resultant force points up.

Magnetic Force on a moving charge

F = BqV

F is the force being undergone by the charge - measured in Newtons (N)

B is the magnetic field strength (or flux density) of the field the charge is in (I.e No. of magnetic lines of force from magnet per unit area) - measured in Teslas (T)

q is the charge of the charge - measured in Coulombs (C).

v is the velocity of the charge - measured in meters per second (ms^-1)

The direction of the Force may be determined using the Right-hand rule.

Orbiting charges

As stated by the Right-hand rule, the direction of the force experienced by a charge entering a magnetic field (assuming the direction of magnetic field vector and the velocity vector of the charge are not the same) would be perpendicular to the plane of the direction of the velocity and magnetic field vectors.

Due to this, as the charge particle's velocity changes due to the force, the force vector also changes. This causes a circular motion of the charge particle inside the magnetic field. Due to this, the charge's path has a radius and a centripetal force. These are described by:

$r={\frac {mv}{Bq}}$

r is the radius of the charge's path - measured in metres (m)

m is the mass of the charge - measured in kilograms (kg)

v is the velocity of the charge - measured in meters per second (ms^-1)

B is the magnetic field strength (or flux density) of the field the charge is in (I.e No. of magnetic lines of force from magnet per unit area) - measured in Teslas (T)

q is the charge of the charge - measured in Coulombs (C).

$F={\frac {mv^{2}}{r}}$

F is the centripetal force being undergone by the charge - measured in Newtons (N)

m is the mass of the charge - measured in kilograms (kg)

v is the velocity of the charge - measured in meters per second (ms^-1)

r is the radius of the charge's path - measured in metres (m)

Parallel Current Carrying Conductors

F/l=k (I₁ I₂)/d

Where F = Force; Newtons N

l = Length of the current carrying conductors parallel to each other.

k = constant; mu0/2p = 2 x 10-7 SI Units

I₁ and I₂ are current carrying conductors respectively

d = distance between the current carrying conductors (mm)

Quick note

Radius is large for more massive, faster particles. Radius is smaller when the magnetic field strength is large