Unit 4 day 2 – Forces on Currents & Charges in Magnetic Fields (B)

• The force exerted on a current carrying conductor by a B-Field

• The Magnetic Force on a Semi-Circular Wire

• Force on an Electric Charge Moving through a B-Field

• Path of an Electric in a B-Field

• A Particle traveling in both an E- & B-Field

The Force Exerted on a Current Carrying Conductor in a B-Field

#2

• Not only does a current in a wire generate a magnetic field and exert a force on a compass needle, but by Newton’s 3rd Law, the reverse is also true. A magnet can also exert a force on a current carrying conductor

Magnetic Force on a Current Carrying Conductor

where l is the length of wire immersed in the magnetic field

• This implies that the direction of the force is perpendicular to the direction of the B-Field (Right Hand Rule #2)

• Then the maximum force is:

BlIF

IlBF

Magnetic Force on a Current Carrying Conductor

• If the direction of the current is not perpendicular to the B-Field, but rather at some angle θ then:

sinIlBBlIF

Magnetic Force on a Current Carrying Conductor

• The equation assumes the magnetic fields is uniform & the current carrying conductor does not make the same angle θ with B

• SI Units for B-Field is Tesla (T) 1 T = 1N/A-m

• We can explore the above equation in differential form:

where dF is the infinitesimal force acting on a differential length dl of the wire

BlIF

BlIdFd

Magnetic Force on a Semi-Circular Wire

IBRF 2

Force on an Electric Charge Moving Through a B-Field

where Δt is the time for charge q to travel a distance l

or the force on a particle is:

tvlandt

NqIwhereBlIF

Btvt

NqF

BvqF

Force on an Electric Charge Moving Through a B-Field

• If , then the force is a maximum and

• If the velocity is at some angle θ wrt the B-Field, then:

Bv qvBF max

sinqvBF

Path of an Electron in a Magnetic Field

• If , then the electron will move in a curved circular path, and the magnetic force acting on it will act like a centripetal force

• The radius of the circular orbit will be:

• The period for 1 revolution will be:

• The Cyclotron Frequency is:

Bv

Bq

vmr

e

e

Bq

mT

e

e2

e

e

m

Bq

Tf

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Path of an Electron in a Magnetic Field

• Note: if the Particle was a proton, the circular path would be upward (counter-clockwise)

A Particle Traveling in Both an E- & B- Field

• The force on a particle traveling in the presence of both an electric and magnetic field which are mutually perpendicular, is given by the Lorentz Equation:

• For a particle to travel straight through, the net force on the particle must equal zero, yielding a velocity selector:

B

Ev

BvEqF