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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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Knowledge Check

  • A current /is flowing through the loop. The direction of the current and the shape of the loop are as shown in the figure. The magnetic field at the centre of the loop is (mu_(0)I)/("R") times (MA=R,MB=2R, angleDMA=90^(@)

    A
    `(5)/(16)` ,but of the plane of the paper.
    B
    `(5)/(16)`, but into the plane of the paper.
    C
    `(7)/(16)`,but out of the plane of the paper.
    D
    `(7)/(16)`,but into the plane of the paper.

  • A wire bent in the shape of a regular n-polygonal loop carries a steady current I. Let l be the perpendicular distance of a given segment and R be the distance of vertex both from the centre of the loop. The magnitude of the magnetic field at the centre of the loop is given by -

    A
    `(n mu_(0)I)/(2pi l)sin(pi//n)`
    B
    `(n mu_(0)I)/(2pi R)sin(pi//n)`
    C
    `(n mu_(0)I)/(2pi l)cos(pi//n)`
    D
    `(n mu_(0)I)/(2pi R)cos(pi//n)`