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Question

A thin glass rod is bent into a semicircle of radius r. A charge +Q is uniformly distributed along the upper half and a charge –Q is uniformly distributed along the lower half, as shown in the figure. The electric field E at, P (the centre of the semicircle, is)

  1. Qπ2ε0r2
  2. 2Qπ2ε0r2
  3. 4Qπ2ε0r2
  4. Q4π2ε0r2


Solution

The correct option is A Qπ2ε0r2
Let us consider a small element dq on the upper half of the semicircular ring at an angle θ with the horizontal.
The electric field due to this element will be dE.
Now consider a similar element in the lower part of the ring and the electric field due to this element will also be dE as shownOn taking the component of this electric field we will get dE cosθas horizontal component and dEsinθ  as vetrical component.
Only vertical components will add up and horzontal component will get cancelled.
dEnet=2dEsinθ........(i)
The electric field dE due to the small element dq will be
dE=kdqr...(ii)
Let dq be at an angle of dθ at the center
dq=Qπr2dθ.......(iii)
Putting (iii) in (ii) we get
dE=2Qdθ4πϵ0r(2)....(iv)
putting (iv) in (i) and integrating we get
dEnet=π20Qsinθdθ(π)2r2ϵ0
Enet=Q(π)2ϵ0r2

 

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