Q.A straight conducting rod AB of resistance R slides smoothly, remaining perpendicular to two long parallel frictionless conducting rails that lie in the plane of the paper and are separated by a perpendicular distance d. A uniform magnetic field of magnitude B, constant in time, is directed out of the paper. The rails have negligible resistance; at their far (left) end they are joined through a capacitor of capacitance C in series with a switch S. The rod is drawn along the rails with a constant velocity v; the switch S is closed at time t = 0 (the capacitor being initially uncharged). Find the current in the sliding rod AB as a function of time.

Motional EMF: Why the Formula Holds

Let's build this from first principles — understanding the why before the formula.

The Core Idea

Motional EMF arises when a conductor moves through a magnetic field. The key insight: moving charges in a magnetic field experience a magnetic force, which acts like a battery pushing charges around the conductor.

Step 1: The Force on a Moving Charge

A charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ feels the Lorentz magnetic force:

This force is perpendicular to both velocity and magnetic field.

Step 2: What Happens Inside a Moving Conductor

Consider a straight metal rod of length $L$ moving with constant velocity $\vec{v}$ perpendicular to a uniform magnetic field $\vec{B}$ (pointing into the page).

• Free electrons in the rod are moving with the rod at velocity $\vec{v}$.
• Each electron experiences a magnetic force:

(negative sign because electron charge is $-e$)

• This force pushes electrons along the rod — say, toward one end.

Step 3: Charge Separation Creates an Electric Field

As electrons accumulate at one end, that end becomes negatively charged, leaving the other end positively charged.

• This charge separation creates an internal electric field $\vec{E}$ inside the rod, pointing from positive to negative end.
• The electric field exerts an opposing force on the electrons:

Step 4: Equilibrium — The "Battery" is Formed

Charge keeps moving until the electric force balances the magnetic force:

Magnitude-wise (for perpendicular $\vec{v}$ and $\vec{B}$):

Step 5: From Electric Field to EMF

The motional EMF $\mathcal{E}$ is the work done per unit charge to move a test charge from one end to the other:

For a uniform field along the rod of length $L$:

The Key Formula

Motional EMF for a straight conductor moving perpendicular to $\vec{B}$:

$\boxed{\mathcal{E} = B L v}$

Why This Makes Physical Sense