An infinitely long hollow cylinder of radiusRcarrying a surface charge densityσis rotated about its cylindrical axis with a constant angular speedω.The magnitude of the surface current isa)2σRωb)σR2ωc)2πσRωd)πσRωCorrect answer is option 'B'. Can you explain this answer?

Question

An infinitely long hollow cylinder of radiusRcarrying a surface charge density&sigma;is rotated about its cylindrical axis with a constant angular speed&omega;.The magnitude of the surface current isa)2&sigma;R&omega;b)&sigma;R2&omega;c)2&pi;&sigma;R&omega;d)&pi;&sigma;R&omega;Correct answer is option 'B'. Can you explain this answer?

Answer

To find the electric field inside the hollow cylinder, we'll use Gauss's law. Gauss's law states that the electric flux through a closed surface is equal to the total charge enclosed divided by the permittivity of free space (ε0).

In this case, we'll consider a Gaussian cylinder of radius r inside the hollow cylinder, with length L. The Gaussian cylinder is chosen such that it is coaxial with the hollow cylinder. Since the hollow cylinder is infinitely long, we can ignore any end effects.

The only charge enclosed by the Gaussian cylinder is the charge on the inner surface of the hollow cylinder. The charge per unit length on the inner surface of the hollow cylinder can be calculated by multiplying the surface charge density (σ) by the circumference of the inner surface of the hollow cylinder (2πR). Therefore, the total charge enclosed is σ(2πR)L.

Using Gauss's law, we can write:

Electric flux = (Total charge enclosed) / (ε0)

The electric flux through the Gaussian cylinder is given by E * (2πrL), where E is the electric field inside the hollow cylinder. Therefore, we have:

E * (2πrL) = σ(2πR)L / ε0

Simplifying, we find:

E = σR / (ε0r)

Thus, the electric field inside the hollow cylinder is given by E = σR / (ε0r).

Answer

Applying Ampere’s law

The correct answer is: σR²ω

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