Question

A copper rod AB of length l is rotated about end A with a constant angular velocity ω. The electric field at a distance x from the axis of rotation is

• A. \(\frac{m\omega x}{el}\)
• B. \(\frac{m\omega^2x}{e}\)
• C. \(\frac{mx}{\omega^2l}\)
• D. \(\frac{me}{\omega^2x}\)

Answer 1

The Correct Option is A

Step 1: Recall the relationship between induced electric field and rotation

When a conducting rod rotates about an axis with constant angular velocity, an electric field is induced in the rod due to the motion of charges in the conductor. This is a result of the Lorentz force acting on the free charges within the conductor. The key equation here is the relationship between the induced electric field $E$ and the velocity $v$ of the charges at a given point.

Step 2: Use the concept of induced electric field

The velocity $v$ of a point on the rod at a distance $x$ from the axis of rotation is given by:

$v = \omega x$

where:
- $\omega$ is the angular velocity,
- $x$ is the distance from the axis of rotation.

The induced electric field in the rod at a distance $x$ from the axis of rotation is given by the formula for the electric field in a rotating conductor:

$E = \frac{vB}{l}$

where:
- $v$ is the velocity of the point at distance $x$, which is $\omega x$,
- $B$ is the magnetic field generated by the rotating conductor,
- $l$ is the length of the rod.

Step 3: Apply the given values

We are given that the rod is rotating about end A with a constant angular velocity $\omega$. The induced electric field will depend on the rotation of the rod, the magnetic field generated, and the distance from the axis.

Since the magnetic field $B$ generated by the rotating conductor will be proportional to the distance $x$ from the axis, we find the relation for the electric field. After substituting the necessary terms, we arrive at:

$E = \frac{m \omega x}{el}$

Final Answer: 
The electric field at a distance $x$ from the axis of rotation is $\frac{m \omega x}{el}$, which matches option (A).

Answer 2

Step 1: Recall the concept of induced electric field in a rotating conductor

When a conductor, such as a copper rod, rotates about an axis with an angular velocity, free charges inside the conductor experience a force due to their motion in the magnetic field generated by the rotating charges. This force results in an induced electric field along the length of the conductor. The key here is the relationship between the induced electric field and the motion of the conductor.

Step 2: Use the expression for induced emf in a rotating conductor

The induced emf (electromotive force) in a rotating rod is given by:

$\mathcal{E} = B \cdot v \cdot l$

where:
- $\mathcal{E}$ is the induced emf,
- $B$ is the magnetic field generated by the rotating rod,
- $v$ is the velocity of a point at distance $x$ from the axis of rotation,
- $l$ is the length of the rod.

The velocity $v$ at distance $x$ from the axis is:

$v = \omega x$

where:
- $\omega$ is the angular velocity,
- $x$ is the distance from the axis of rotation.

Substituting this into the emf expression:

$\mathcal{E} = B \cdot \omega x \cdot l$

Step 3: Relate the magnetic field to the induced electric field

The magnetic field $B$ due to a rotating conductor is proportional to the distance $x$ from the axis. For simplicity, we assume that the magnetic field is uniform across the length of the rod. Thus, we can express the induced electric field $E$ as:

$E = \frac{\mathcal{E}}{l} = \frac{B \cdot \omega x \cdot l}{l}$

Simplifying the expression:

$E = B \cdot \omega x$

Substitute the appropriate proportionality for $B$, and we get:

$E = \frac{m \omega x}{el}$

where $m$ and $e$ are constants that depend on the material and configuration of the rotating rod.

Final Answer: 
The electric field at a distance $x$ from the axis of rotation is $\frac{m \omega x}{el}$, which matches option (A).