A loop of wire in the shape of a rectangle of width w and length l, and a long s

Question

A loop of wire in the shape of a rectangle of width w and length l, and a long straight wire carrying a current I are in a plane as shown. [Note: express all of your final answers in terms of l, w, r (defined in the figure), I_2 (defined below), numerical constants (such as pi), and constants that appear in Maxwell's equations.]

THe questions are about the figure below.

1) What is the magnitude and direction of the magnetic field at the exact center of the loop?

2) What is the magnetic flux through the loop due to current I? (Hint: you can't just multiply by an area, do an integral)?

3) If a current I_2 runs clockwise around the loop, what is the net force on the loop?

The Answers are as follows:

1) (m_u * I) / (2pi (r + w/2)) into the page

2) (m_u*I*l) / (2pi) * ln (1 + w/r)

3) (m_u*I*I_2*l) / (2pi) * (1/r - 1/(r+w) ) to the left

Points will only be awarded to a COMPLETE explanation to all these answers and their directions, thanks.

Explanation / Answer

A) MF at distance r = (Uo)i/2pi(r) and direction is given by right ahnd rule....at centre r =r+w/2 and direction will be into the page

so B = (Uo)I/(2pi)(r+(w/2))

B) at distance x take element of dx from wire...so elemental area dA = L(dx)

at this point the MF is approx constant...so d(flux) = B dA = (Uo)I/((2pi)x) L dx

so flux = integral (Uo)i/((2pi)x) L dx from x =r to x= r+w

hence using integral dx/x = ln(x) +c

we get flux = (Uo*I*L) / (2pI) ln (1 + w/r)

C) due to symmetry the forces on upper and lower branches cancel each other...

so net force will be due to left and right side of branches/..

force on wire of length L in B ...BIL and direction is given by left hand rule...

so net force = (Uo)i/2pi(r) L *I_2 - I_2* L *(Uo)i/2pi(r+w) towards wire.. = L *I_2 *(Uo)i/2pi [1/r - 1/r+w]

where Uo = 4 pi *10^-7

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