A current I travels counterclockwise through a closed copper wire loop which has the shape of a rectangle with sides 'a' and 'b'.
What is the magnitude of the magnetic field at the center, C, of the rectangle? Express your answer in terms of a, b, I and mu _0. (Enter mu_0 )
To find the magnitude of the magnetic field at the center of the rectangle, we can make use of Ampere's Law. Ampere's Law relates the magnetic field around a closed loop to the current passing through the loop.
1. Start by drawing a rectangular loop with sides 'a' and 'b'. Label the center of the rectangle as point C.
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a b
2. Apply Ampere's Law, which states that the integral of the magnetic field, B, dot the elemental length, dl, around a closed loop is equal to the product of the current enclosed, I_enc, and the permeability of free space, mu_0:
∮ B · dl = mu_0 * I_enc
3. Considering point C at the center of the rectangle, the magnetic field will have the same magnitude and direction at every point on a circular path centered at C, with a radius equal to half the length of the shortest side of the rectangle. In this case, that would be b/2.
4. So, draw a circle centered at C, with a radius equal to b/2.
5. Now we need to find the current enclosed by this circular path. Since the current is flowing counterclockwise through the entire loop, both horizontal sides of length 'a' will contribute to this enclosed current.
6. Calculate the current enclosed:
I_enc = I * 2
7. Plug in the values into Ampere's Law:
∮ B · dl = mu_0 * I_enc
∮ B · dl = mu_0 * (I * 2)
8. Simplify the equation and solve for the magnetic field:
B ∮ dl = 2mu_0 * I
9. The magnetic field is constant around the circular path, so we can take it out of the integral:
B * (2π * (b/2)) = 2mu_0 * I
10. Simplify the equation:
B * π * b = 2mu_0 * I
11. Solve for the magnetic field, B:
B = (2mu_0 * I) / (π * b)
Therefore, the magnitude of the magnetic field at the center of the rectangle, C, is given by:
B = (2mu_0 * I) / (π * b)
Hence, the answer is:
B = (2mu_0 * I) / (π * b)
And mu_0 represents the permeability of free space.