A long, cylindrical conductor of radius R carries a current I as shown in the figure below. The current density J, however, is not uniform over the cross-section of the conductor but is a function of the radius according to J = 6br^4, where b is a constant.
Find an expression for the magnetic field magnitude B at the following distances, measured from the axis. (Use any variable or symbol stated above along with the following as necessary: μ0.)
(a) r1 < R
B = ?
(b) r2 > R
B = ?
To find the expression for the magnetic field magnitude at the given distances, we can use Ampere's Law, which relates the magnetic field around a closed loop to the current passing through it.
Ampere's Law states that the line integral of the magnetic field B around a closed loop is equal to μ0 times the total enclosed current I_enc:
∮ B · dl = μ0 * I_enc
In this case, we can consider a circular loop centered on the axis of the cylinder with radius r. The current passing through this loop will be the sum of all the currents contained within the loop at that radius.
(a) r1 < R
For r1 < R, the complete current is enclosed within the loop. Thus, the total enclosed current I_enc is equal to the total current I.
Using Ampere's Law, we have:
∮ B · dl = μ0 * I
The left side of the equation represents the line integral of the magnetic field B along the circumference of the loop. For a circular loop of radius r1, the line integral simplifies to B * 2πr1.
Therefore, we have:
B * 2πr1 = μ0 * I
Simplifying for B, we get:
B = (μ0 * I) / (2πr1)
(b) r2 > R
For r2 > R, part of the conductor lies outside the loop. Therefore, the total enclosed current I_enc will be less than the total current I.
In this case, we need to calculate the current enclosed within the loop at radius r2. We can do this by integrating the current density J over the cross-sectional area of the loop.
To find the current enclosed within the loop at radius r2, we can integrate the current density function J = 6br^4 over the cross-sectional area of the loop.
The cross-sectional area of the loop is given by A = πr^2, where r is the radius of the loop.
Therefore, the current enclosed within the loop at radius r2 is:
I_enc = ∫ J dA
= ∫ (6br^4) dA
= 6b ∫ r^4 dA
= 6b ∫ r^4 * 2πr dr (integrated over the limits of r1 to r2)
Simplifying this integral, we get:
I_enc = 12πb ∫ r^5 dr (integrated over the limits of r1 to r2)
Evaluating this integral, we get:
I_enc = 12πb * [(r2^6)/6 - (r1^6)/6]
= 2πb [r2^6 - r1^6]
Using Ampere's Law, as mentioned before:
∮ B · dl = μ0 * I_enc
The line integral of B along the circumference of the loop for r2 > R simplifies to B * 2πr2.
Therefore, we have:
B * 2πr2 = μ0 * I_enc
Simplifying for B, we get:
B = (μ0 * I_enc) / (2πr2)
B = (μ0 * 2πb [r2^6 - r1^6]) / (2πr2)
B = (μ0 * b * (r2^6 - r1^6)) / r2
So, the expression for the magnetic field magnitude at distance r1 < R is (μ0 * I) / (2πr1), and at distance r2 > R is (μ0 * b * (r2^6 - r1^6)) / r2.