A A current I flows in the direction indicated in a wire loo

A. A current I flows in the direction indicated in a wire loop lying in the x-y plane as shown. Use the Biot-Savart law to derive the magnetic field B at the centre of the circle. (Hint: Derive the contribution due to each segment of the current loop, i.e. a b, b c, c d and d a.)

Solution

For any small length, say dl, along the curve, the magnetic field at the centre is given as\"

dB = µo I dl/ 4R^2 and the direction of this magnetic field would be directed outwards from the page. [Curl your fingers around the wire with thumb along the direction of current. The curl of fingers give the direction of current]

or, B = dB = µo I dl/ 4R^2 = 3 µo I (2R)/ 16R^2 = 3 µo I / 8R

Again, for the straight wire section:

We have a current I flowing between b and c, Here angle Rbc and angle Rcb would be 45 degrees.

Also, we have B(due to straight line) = (µoI 2 / 8R) [Cos1 + Cos2] [Here 1 and 2 are the angles subtended at the two ends of the wire]

Hence B(due to straight line) = (µoI 2/ 8R) [2] = µoI/ 4R and the direction of the current would again be out of the page.

Therefore, net magnetic field at the centre = B(due to curve) + B(due to straight line) [The magnetic field due to the straight sections ab and cd would be zero as the direction of the flow of the current is towards the centre itself]

Net magnetic field = 3 µo I / 8R + µoI/ 4R = (µoI/R)[3/8 + 1 /4]

NOTE: For such questions, always try to find the magnetic field due to each of the components(for which the field can be easily determined) and then put them up as per the direction of the magnetic fields to find the net value.