A long coaxial cable consists of two concentric conductors. The inner conductor is a cylinder with radius R_1, and it carries a current I_0 uniformly distributed over its cross section. The outer conductor is a cylindrical shell with inner radius R_2 and outer radius R_3. It carries a current I_0 that is also uniformly distributed over its cross section, and that is opposite in direction to the current of the inner conductor. Calculate the magnetic field {B} as a function of the distance R from the axis.
What is the direction and magnitude of the magnetic field for 0<r<R_1? Express your answer in terms of I_0, R_1, R_2, R_3, r and \mu _0 (enter mu_0 for \mu _0)
What is the direction and magnitude of the magnetic field for R_1<r<R_2? Express your answer in terms of I_0, R_1, R_2, R_3, r and \mu _0 (enter mu_0 for \mu _0)
What is the direction and magnitude of the magnetic field for R_2<r<R_3? Express your answer in terms of I_0, R_1, R_2, R_3, r and \mu _0 (enter mu_0 for \mu _0)
Use the Biot-Savart law: magnetic field do to an enclosed current.
a. enclosed current is proportonal to enclosed area, or 1/r^2
B=mu*Ienclosed/2PIr
but I enclosed=Io*r /R1^2
ans: B=mu*Io*r /R1^2
b. enclosed current is Io
B=mu*Io/2PI r
c. do the same as A, but you have to add (b) to it.
Enclosed current=Io-Io(r-R2)^2/(r-R3)^2
B=mu*Io(1-(r-R2)^2/(r-R3)^2 * 1/2PIr
check that.
a) and c) are wrong
part a) and c)are wrong !
To calculate the magnetic field as a function of distance R, we need to apply Ampere's Law to each region of the coaxial cable.
1. For 0 < r < R_1:
- The magnetic field inside the inner conductor is given by the formula: B = (mu_0 * I_0) / (2 * pi * r), where mu_0 is the permeability of free space.
- The direction of the magnetic field is determined by the right-hand rule, with the thumb pointing in the direction of the current in the inner conductor. The magnetic field lines will form concentric circles around the axis.
2. For R_1 < r < R_2:
- The magnetic field between the inner and outer conductors is constant and given by the formula: B = (mu_0 * I_0) / (2 * pi * R_1), since the current is uniformly distributed.
- The direction of the magnetic field is radially outward, from the axis of the cable toward the outer conductor.
3. For R_2 < r < R_3:
- The magnetic field inside the outer conductor is given by the formula: B = (mu_0 * I_0) / (2 * pi * r), where mu_0 is the permeability of free space.
- The direction of the magnetic field is determined by the right-hand rule, with the thumb pointing in the direction opposite to the current in the outer conductor. The magnetic field lines will form concentric circles around the axis.
It's important to note that the direction of the magnetic field can be inferred using the right-hand rule, while the magnitude is determined by the formulas mentioned above, depending on the region of interest.