Question

A thick wire of radius R carries a constant current I (see figure below). Find the magnetic field inside the wire, a distance r from the center of the wire and far from the ends of the wire, where r < R and R is much less than the length of the wire. (Enter the magnitude. Use any variable or symbol stated above along with the following as necessary: μ0.)

Answer 1

The magnetic field inside a thick wire at a distance r from the center can be found by applying Ampère's Law, which ultimately gives B = (µ_0 I / 2πR^2) ⋅ r as the magnetic field's magnitude at that point.

Step-by-step explanation:

To find the magnetic field inside a thick wire at a distance r from the center, we apply Ampère's Law, which relates the magnetic field along a closed loop to the current enclosed by that loop. Considering a cylindrical wire of radius R, carrying a current I, and a point inside at a distance r (< R), we can use Ampère's Law in the form:

µ0 Ienc = B ⋅ (2πr)

Here, Ienc is the current enclosed by the loop that corresponds to our point at distance r from the center. Since the current density J is constant across the wire's cross-section, we can say that the current enclosed is a fraction of the total current I, proportional to the area of the circle at radius r compared to the total cross-sectional area of the wire:

Ienc = I ⋅ (πr2 / πR2) = I ⋅ (r2 / R2)

Plugging this expression into Ampère's Law, we get the magnetic field B at distance r:

B ⋅ (2πr) = µ0 I ⋅ (r2 / R2)

Therefore:

B = (µ0 I / 2πR2) ⋅ r

This gives us the magnetic field's magnitude inside the wire.