A wire of radius R carries a non-uniform current: the current density, j, (current per unit area) varies as a function of distance r from the central axis of the wire according to j = j0(R/r) (and j = 0 outside the wire, of course). Find expressions for the strength of the magnetic field in the regions 0 < r < R and R < r.
Use Ampère's law:
http://en.wikipedia.org/wiki/Amp%C3%A8re's_circuital_law
2 pi r B(r) =
Integral from r' = 0 to r of
2 pi r' j(r') dr
Typo:
2 pi r B(r) =
µ_0 Integral from r' = 0 to r of
2 pi r' j(r') dr
To find the expressions for the strength of the magnetic field in the regions 0 < r < R and R < r, we can use Ampere's Law. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the permeability of free space.
Let's consider a circular loop with radius r and apply Ampere's Law to find the magnetic field.
1. For 0 < r < R (inside the wire):
We need to find the magnetic field inside the wire, where the current density varies as j = j0(R/r).
- Choose a circular loop of radius r inside the wire. The current passing through this loop is given by I = j * A, where A is the area of the loop.
- Since the current density is proportional to (R/r), we can rewrite the current passing through the loop as I = j0(R/r) * A.
- The line integral of the magnetic field B around the loop is equal to B * 2πr, where 2πr is the circumference of the loop.
- Applying Ampere's Law, we have B * 2πr = μ0 * I, where μ0 is the permeability of free space.
By substituting the expression for I, we get B * 2πr = μ0 * j0(R/r) * A.
The area of the loop is A = πr^2, so we have B * 2πr = μ0 * j0(R/r) * πr^2.
Solving for B, we find the expression for the magnetic field inside the wire as:
B = (μ0 * j0 * R^2) / (2r), where 0 < r < R.
2. For R < r (outside the wire):
In this region, the current density is zero since there is no current outside the wire.
- Choose a circular loop of radius r outside the wire. The current passing through this loop is zero, so the line integral of the magnetic field B around the loop is also zero.
- Applying Ampere's Law, we have B * 2πr = 0.
Thus, the magnetic field in this region is zero: B = 0, where R < r.
To summarize:
- Inside the wire (0 < r < R), the magnetic field strength is B = (μ0 * j0 * R^2) / (2r).
- Outside the wire (R < r), the magnetic field strength is B = 0.