Motion of a Straight Conductor in a Uniform Magnetic Field (Motional EMF)

When a straight conductor of length l moves with velocity v perpendicular to a magnetic field B, an emf or potential difference is developed between its ends. This is called motional emf.

It is called motional emf because it is induced due to the motion of the conductor.

Consider a thin conducting rod JK of length l:

• The rod is placed in a uniform magnetic field \[\vec B\].
• The magnetic field is perpendicular to the plane of the paper and directed downwards into the paper.
• The field is represented by crosses (×).
• The rod moves in the plane of the paper, perpendicular to its own axis.
• The rod has constant velocity \[\vec v\] towards the right.

A conductor contains free electrons and positive charges.

• The free electrons can move inside the conductor.
• When the rod moves rightward, the positive and negative charges move with the rod in the magnetic field.
• A charge q, moving with velocity \[\vec v\] perpendicular to the magnetic field \[\vec B\], experiences magnetic force \[\vec F_m\].
  F_m = qvB
  This magnetic force is called the Lorentz force.
• Its direction is perpendicular to both \[\vec B\] and \[\vec v\].
• According to Fleming’s left-hand rule, the force on a positive charge is towards J.
• The force on a negative charge is towards K.

Since electrons are free to move, they move towards end K.

• End J has a deficiency of electrons and becomes positively charged.
• End K has an excess of electrons and becomes negatively charged.
• Hence, an electric potential difference V is induced between the ends of the rod.

Result: J becomes positive, and K becomes negative.

Due to charge separation, an electric field \[\vec E\] is created inside the rod.

• The electric field is directed from J (positive end) to K (negative end).
• This electric field exerts an electric force \[\vec F_e\]​ on each charge q
  F_e = qE
• For a positive charge, the electric force is towards K.
• For a negative charge, the electric force is towards J.
• Therefore, electric force is opposite to magnetic force.

As more electrons reach end K, the electric force increases.

Finally, the electric force becomes equal to the magnetic force:

From equation (i): E = \[\frac {V}{l}\]

Comparing with equation (ii): \[\frac {V}{l}\] = vB

where B is the magnetic field, v is the velocity of the conductor, and l is the length of the conductor.

If:

• B is in weber per metre squared,
• v is in metres per second,
• l is in metres,

then V is obtained in volts.

1. Train axle: A train axle moving along rails may cut components of Earth’s magnetic field. A small potential difference can develop between its ends.
2. Aircraft wings: An aeroplane moving through Earth’s magnetic field can develop a potential difference between its wing tips, depending on the orientation of the wings and direction of motion.
3. Sliding rod on rails: A rod sliding on conducting rails in a magnetic field is a standard laboratory and examination model of motional emf. If the circuit is closed, induced current flows.

• If the conductor moves with velocity \[\vec v\] at an angle other than 90° with magnetic field \[\vec B\], then the induced potential difference is:
  V = \[\vec E\] . \[\vec l\] = (\[\vec v\] × \[\vec B\]) . \[\vec l\]
• If either velocity \[\vec v\] or conductor length \[\vec l\] is parallel to magnetic field \[\vec B\], no potential difference develops between the ends of the conductor.
• Fleming’s right-hand rule can be used to determine the direction of induced current in a straight conductor.
• The horizontal component of Earth’s magnetic field is directed from south to north.
• The vertical component of Earth’s magnetic field is directed vertically downwards.