Question

A solid cylinder carries a current that is uniform over its cross section. Where is the magnitude of the magnetic field the greatest? a) at the center of the cylinder's cross section b) in the middle of the cylinder c) at the surface d) none of the above

Short answer

Answer: c) at the surface

Step-by-step solution

Ampère's Law

Ampère's Law states that the integral of the magnetic field B around a closed loop (or Amperian loop) is equal to the product of the permeability of free space (μ₀) and the total current passing through the loop. Mathematically, it is expressed as: ∮B⃗ · d⃗l = μ₀ * I_enclosed In cylindrical coordinates, the magnetic field generated by the current in the cylinder will have only the azimuthal component.

Consider points inside the cylinder

We need to evaluate the magnetic field inside the cylinder at a radial distance r from the central axis where 0 ≤ r ≤ R, with R being the radius of the cylinder. The field inside the cylinder depends on the portion of the current enclosed inside the Amperian loop. The Amperian loop will be a circle with radius r inside the cylinder. We need to find the enclosed current inside this loop. Let the current per unit area be J (in A/m²). Then, the total current flowing through the cylinder can be found by integrating J over the cylinder's cross-sectional area. The enclosed current within the Amperian loop is: I_enclosed = J * π * r²

Use Ampère's Law for points inside the cylinder

Now, using Ampère's Law for points inside the cylinder: ∮B⃗ · d⃗l = B * 2 * π * r = μ₀ * J * π * r² Solving for B, we have: B_inside = (μ₀ * J * r) / 2

Consider points outside the cylinder

For points outside the cylinder (i.e., r > R), the entire current is enclosed within the Amperian loop. The current enclosed is then: I_enclosed = J * π * R²

Use Ampère's Law for points outside the cylinder

Using Ampère's Law for points outside the cylinder: ∮B⃗ · d⃗l = B * 2 * π * r = μ₀ * J * π * R² Solving for B, we have: B_outside = (μ₀ * J * R²) / (2 * r)

Compare the magnetic field magnitudes

Now we can compare the magnetic field's magnitudes depending on the radial distance r: - At the center of the cylinder's cross section (r = 0), the magnetic field B_inside = 0. - In the middle of the cylinder (0 < r < R), the magnetic field B_inside is proportional to r. - At the surface (r = R), B_outside = (μ₀ * J * R) / 2. - For points outside the cylinder (r > R), the magnetic field B_outside decreases with increasing r.

Determine the location of the greatest magnitude

Based on the comparison, we can see that as we move from the center to the surface, the magnetic field's magnitude increases from 0 to a maximum value (μ₀ * J * R) / 2 at the surface, and then decreases as we move further away from the surface. Therefore, the magnitude of the magnetic field is the greatest at the surface of the cylinder. The correct answer is: c) at the surface