Question

A conducting disc rotates about its axis (passing through centre and perpendicular to the plane of disc) in a uniform magnetic field \(\mathrm{B}\) and magnetic field is perpendicular to plane of disc. There are certain points marked on a chord joining two points \(\mathrm{C}\) and G on the periphery of the disc.Point \(E\) is the midpoint of the chord. Points \(D\) and \(\mathrm{F}\) are the midpoints of segments \(\mathrm{CE}\) and EG respectively. The emf induced is maximum between the points:(Do not consider the emf due to centrifugal forces) (1) \(C\) and \(D\) (2) \(\mathrm{C}\) and \(\mathrm{E}\) (3) \(\mathrm{C}\) and \(\mathrm{G}\) (4) \(\mathrm{D}\) and \(\mathrm{F}\)

Short answer

The emf induced is maximum between points C and G.

Step-by-step solution

- Understanding the rotation and magnetic field

The conducting disc is rotating in a uniform magnetic field which is perpendicular to the plane of the disc. This setup will have an electromotive force (emf) induced in different parts of the disc due to the changing magnetic flux through different sections as a function of their distance from the rotation axis.

- Apply Faraday's Law of Induction

According to Faraday's Law, the emf induced in a conductor is directly proportional to the rate of change of magnetic flux through the conductor. For a rotating disc, the induced emf is also dependent on the radius of the disc and the angular velocity (which is constant here). The emf is maximum where the rate of magnetic flux change is highest.

- Identify relative positions of points

Check the positions of points C, D, E, F, and G on the disc. Point C and G are on the periphery, while E is the midpoint of chord CG. D and F are midpoints of segments CE and EG, respectively. This forms the sequence C-D-E-F-G.

- Evaluate the segments for maximum emf

To have the maximum emf induced, the distance between the points should ideally cover the maximum possible part of the disc because more area coverage means more flux change. Segment CG spans the entire chord length which is the maximum distance between two points considered here.

- Conclusion

Between the given options, segment CG covers the maximum area affected by the magnetic flux change, hence it will have the maximum induced emf.

Key concepts

Faraday's Law of Induction

Faraday's Law of Induction is a fundamental principle in electromagnetism discovees by Michael Faraday. It states that the electromotive force (EMF) induced in a circuit is directly proportional to the rate of change of magnetic flux through that circuit. This can be mathematically expressed as:
\[ \mathcal{E} = - \dfrac{d \Phi}{dt} \]
Here, \( \mathcal{E} \) represents the induced EMF, and \( \Phi \) is the magnetic flux. In the context of a rotating disc in a magnetic field:

• As the disc rotates, different parts of it move through varying amounts of magnetic flux.
• Changes in magnetic flux lead to the generation of EMF across the disc's various sections.

The negative sign in the equation signifies the direction of the induced EMF, opposing the change in magnetic flux (Lenz's Law).

Electromotive Force (EMF)

Electromotive Force, commonly known as EMF, is the electrical action produced by a non-electrical source. It can be thought of as the potential difference across a source when no current flows. The unit of EMF is volts (V).
In a rotating disc under a uniform magnetic field, EMF is generated due to the rotation:

• Maximum EMF is observed where there is a higher rate of flux change.
• In our exercise scenario, the maximum EMF is induced between points that span the largest area of the disc experiencing the magnetic field change.

Remember, EMF helps to move charges within the conductor, initiating an electric current when there’s a closed loop.

Magnetic Flux

Magnetic flux, denoted by \( \Phi \), represents the quantity of magnetic field passing through a given area. It's measured in Webers (Wb). The formula for magnetic flux is:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
Where

• \( B \) is the magnetic field strength
• \( A \) is the area perpendicular to the magnetic field
• \( \theta \) is the angle between the magnetic field lines and the perpendicular to the surface.

For a rotating disc in a uniform magnetic field:

• The flux changes as different parts of the disc rotate through the magnetic field.
• More area covered by the segment of the disc (like CG) undergoes more flux change, leading to higher induced EMF.

Understanding the magnetic flux gives insight into why certain points on the disc experience a higher rate of change, and thus, higher induced EMF.