The magnetic field at any point in the xy plane is given by , where is the position vector of the point, A is a constant, and is a unit vector in the +z direction.The net current through a circle of radius R, in the xy plane and centered at the origin is given by: A)" $\pi$ AR²/ $\mu$ ₀" B)"2 $\pi$ AR/ $\mu$ ₀" C)"4 $\pi$ AR³/3 $\mu$ ₀" D)"2 $\pi$ AR²/ $\mu$ ₀" E)" $\pi$ AR²/2 $\mu$ ₀"

Question

Quizplus · Accepted Answer

The net current passing through the circle of radius R and centered at origin is given by the line integral of the magnetic field  around the circle, which can be expressed as:

∮B⋅dℓ=∮(A/R)ρz^⋅dℓ, where ρ = R (since the circle is in the xy plane and centered at origin)

By considering a parametric form of the circle as x = Rcosθ, y = Rsinθ, and z = 0, and taking the differential of this equation to get the differential vector, we can evaluate the dot product and write the line integral as:

∮B⋅dℓ=∫0^2π(A/R)Rcosθ(z^⋅dℓx+0+0)+Rsinθ(z^⋅dℓy+0+0)dθ

Now, z^⋅dℓx = 0 and z^⋅dℓy = 0, because the line element dℓ is tangent to the circle and hence perpendicular to the z-axis. Thus,

∮B⋅dℓ=∫0^2π0dθ=0

Therefore, the net current passing through the circle is zero, and the correct answer is D.

The Magnetic Field at Any Point in the Xy Plane π\piπ

The Magnetic Field at Any Point in the Xy Plane $\pi$