A long thick conducting cylinder of radius‘R’carries a current uniformly distributed over its cross section.a)The strength of the magnetic field inside the cylinder will vary as inversely proportional tor, whereris the distance from the axis.b)The energy density of the magnetic field outside the conductor varies as inversely proportional to1/r2, where‘r’ is the distance from the axis.c)The magnetic field strength is maximum on the surfaced)The magnetic field strength is zero on the surfaceCorrect answer is option 'B,C'. Can you explain this answer?

Question

A long thick conducting cylinder of radius&lsquo;R&rsquo;carries a current uniformly distributed over its cross section.a)The strength of the magnetic field inside the cylinder will vary as inversely proportional tor, whereris the distance from the axis.b)The energy density of the magnetic field outside the conductor varies as inversely proportional to1/r2, where&lsquo;r&rsquo; is the distance from the axis.c)The magnetic field strength is maximum on the surfaced)The magnetic field strength is zero on the surfaceCorrect answer is option 'B,C'. Can you explain this answer?

Answer

R and length L is oriented along the z-axis. The cylinder has a uniform charge density ρ.

To find the electric field inside and outside the cylinder, we can 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 electric constant ε0.

Let's first consider a Gaussian surface in the shape of a cylinder with radius r1 and length L, placed inside the conducting cylinder. The electric field inside the conducting cylinder is radially symmetric, so we can choose the Gaussian surface as a cylinder with its axis along the z-axis. The electric field is constant on the surface of the Gaussian cylinder, and its magnitude is E1.

The electric flux through the Gaussian surface is given by Φ1 = E1 * A1, where A1 is the surface area of the Gaussian cylinder. The charge enclosed by the Gaussian surface is given by Qenc = ρ * V, where V is the volume of the Gaussian cylinder.

According to Gauss's Law, Φ1 = Qenc / ε0.

Since the electric field inside the conductor is zero, the charge enclosed by the Gaussian surface is also zero. Therefore, Qenc = 0, and the electric flux through the Gaussian surface is also zero. This implies that E1 * A1 = 0, which means that E1 = 0.

Hence, the electric field inside the conducting cylinder is zero.

Now, let's consider a Gaussian surface in the shape of a cylinder with radius r2 and length L, placed outside the conducting cylinder. The electric field outside the conducting cylinder is also radially symmetric, so we can choose the Gaussian surface as a cylinder with its axis along the z-axis. The electric field is constant on the surface of the Gaussian cylinder, and its magnitude is E2.

The electric flux through the Gaussian surface is given by Φ2 = E2 * A2, where A2 is the surface area of the Gaussian cylinder. The charge enclosed by the Gaussian surface is given by Qenc = ρ * V, where V is the volume of the Gaussian cylinder.

According to Gauss's Law, Φ2 = Qenc / ε0.

The charge enclosed by the Gaussian surface is equal to the total charge of the conducting cylinder. Therefore, Qenc = ρ * π * r^2 * L.

Substituting these values into Gauss's Law, we get Φ2 = (ρ * π * r^2 * L) / ε0 = E2 * (2 * π * r2 * L).

Simplifying this equation, we find E2 = (ρ * r^2) / (2 * ε0 * r2).

Hence, the electric field outside the conducting cylinder is given by E2 = (ρ * r^2) / (2 * ε0 * r2).

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