A semicircular loop of radius a carries positive charge Qdistributed uniformly.(

Question

A semicircular loop of radius a carries positive charge Qdistributed uniformly.(Figure 1) Find the electric field at the loop's center (point P in the figure). Hint: Divide the loop into charge elements dq as shown in the figure, and write dq in terms of the angle d?, then integrate over ?. Express your answer in terms of i^, j^, k, Q, a. E? =______________

A semicircular loop of radius a carries positive charge Qdistributed uniformly.(Figure 1) Find the electric field at the loop's center (point P in the figure). Hint: Divide the loop into charge elements dq as shown in the figure, and write dq in terms of the angle d?, then integrate over ?. Express your answer in terms of i^, j^, k, Q, a. E? =

Explanation / Answer

The field dE is defines as k*dq/r^2 but due to symmetry only the downward component is not canceled

So dEy = k*dq*sin(?)/r^2 but dq = ?*r*d? where ? (is the charge density) = Q/?*r so dq = Q/?*d?

so dEy = k*Q/?/r^2*sin(?)*d? here r =a and the limits of integration are 0 to ?

so the result Ey = k*Q/?/a^2*(-cos(?) - -cos(0)) = k*Q/?/a^2*2

so Ey = 2*k*Q/(?*a^2)

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