A cylindrical conductor of radius carries a uniformly distributed current I. Show that the magnetic field within the conductor measured at a distance r from the center is given by the following equation...
B= (u0Ir)/2pi a^2
Use either Biot Savart Law, or Amperes law: the magnetic field is related to the current enclosed....
To determine the magnetic field within the cylindrical conductor, 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 (μ0).
Let's consider a circular Amperian loop of radius r within the cylindrical conductor. The magnetic field will be constant along the loop due to the uniform distribution of current. The symmetry of the problem implies that the magnetic field will be directed tangent to the circular loop and will be purely radial.
The next step is to calculate the magnetic field using Ampere's Law. The equation for Ampere's Law is given as:
∮ B · dl = μ0 * Iin
where B is the magnetic field, dl is a small differential segment along the loop, and Iin is the current passing through the Amperian loop.
Since the magnetic field is radial, we can rewrite the equation as:
∫ B * dl = μ0 * Iin
Now, we need to evaluate the left-hand side of the equation. The line integral of the magnetic field (B * dl) along the circular loop is equal to B multiplied by the circumference of the loop (2πr), as the magnetic field is constant along the loop:
B * 2πr = μ0 * Iin
Rearranging the equation, we get:
B = (μ0 * Iin) / (2πr)
Since the current is uniformly distributed across the conductor, the current passing through the Amperian loop (Iin) will be the total current (I) multiplied by the ratio of the area of the Amperian loop to the total area of the conductor (A):
Iin = I * (Ain / A)
The area of the Amperian loop (Ain) is given by the formula for the area of a circle:
Ain = πr^2
The total area of the conductor (A) is also given by the formula for the area of a circle:
A = πa^2
Substituting the values of Iin, Ain, and A into our equation for the magnetic field, we get:
B = (μ0 * I * (Ain / A)) / (2πr)
B = (μ0 * I * (πr^2) / (πa^2)) / (2πr)
Simplifying, we get:
B = (μ0 * Ir) / (2πa^2)
Therefore, the magnetic field within the cylindrical conductor, measured at a distance r from the center, is given by B = (μ0 * Ir) / (2πa^2), where B is the magnetic field, I is the current, r is the distance from the center of the conductor, a is the radius of the conductor, and μ0 is the permeability of free space.