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  4. The current in flowing along the path ABCD of a cube (shown in the left figure) produces a magnetic field at the centre of cube of magnitude B. Dashed line depicts the non-conducting part of the cube. <img class=img-fluid question-image alt=image src=https://cdn.tardigrade.in/img/question/physics/5f14c22d2ce4c2b1ae1feb7d3610c350-.jpeg /> Consider a cubical shape shown to the right which is identical in size and shape to the left. If the same current now flows in along the path DAEFGCD, then the magnitude of magnetic field at the ecnntre will be

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Q. The current in flowing along the path ABCD of a cube (shown in the left figure) produces a magnetic field at the centre of cube of magnitude B. Dashed line depicts the non-conducting part of the cube.
image
Consider a cubical shape shown to the right which is identical in size and shape to the left. If the same current now flows in along the path $DAEFGCD,$ then the magnitude of magnetic field at the ecnntre will be

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Solution:

Given magnetic field due to a square loop $DABCA$ at centre of cube is $B.$
image
Now, loop $DAEFGCD$ can be viewed as super position of three square loops as shown below.
image
So, net field at centre = resultant of fields of these three loops
image
$\therefore B_{ net }=B \hat{i}+B\hat {j}-B \hat{k}$
Magnitude of resultant field is
$|B_{net}| =\sqrt{B^{2}+B^{2}+B^{2}=}\sqrt{3}.B$