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A steady current I goes through a wire l...

A steady current `I` goes through a wire loop `PQR` having shape of a right angle triangle with `PQ = 3x, PR = 4x and QR = 5x`. If the magnitude of the magnetic field at `P` due to this loop is `k((mu_(0)I)/(48 pi x))`, find the value of `K`.

A

1

B

5

C

10

D

7

Text Solution

Verified by Experts

The correct Answer is:
D
Using the concept of area of triangle
`1/2 xx PD xx 5x = 1/2 xx 3x xx 4x`
`therefore PD= (12x)/5`

`QD = sqrt((PQ)^(2) - (PD)^(2)) =sqrt(9x^(2) - (144x^(2))/(25)) = (9x)/5`
and `DR = 5x - (9x)/5 = (16x)/5`
Magnetic field at P due to current elements PQ and PR is zero as the point P is on the conductor. Therefore, magnetic field at P due to current element QR is
`B=(mu_(0)I)/(4pi PD)(sin phi_(1) + sin phi_(2))`
`B=(mu_(0)I xx 5)/(4pi xx 12x)((9x//5)/(3x) + (16x//5)/(4x))`
`=(mu_(0)I5)/(48 pi x)(3/5 + 4/5) = (7mu_(0)I)/(48 pi x) therefore k = 7`
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