It's still zero!

A circular wire loop of radius R R carries a total charge q q distributed uniformly over its length. A small length x < < R x<<R of the wire is cut off. Find the electric field at the centre due to the remaining wire.

If the answer is in the form of q x α π β ε 0 R δ \dfrac { qx }{ \alpha { \pi }^{ \beta }{ \varepsilon }_{ 0 }{ R }^{ \delta } } , find α + β + δ \alpha +\beta +\delta .

Assume the medium to be vacuum.

Clarification: ε 0 { \varepsilon }_{ 0 } is the permittivity of free space.


The answer is 13.

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2 solutions

Rishabh Jain
Mar 16, 2016

Let C R CR denote complete ring, C O CO represents cutout and R R RR denotes remaining ring. Then by principle of superposition:- E R R + E C O = E C R = 0 E_{RR}+E_{CO}=E_{CR}=0 Since Electric field due to complete ring at centre is 0 0 . E R R = E C O E_{RR}=-E_{CO} Now since cut out is very small it can be treated as a point charge on which charge is given by ( q 2 π R ) × x \left(\dfrac{q}{2\pi R}\right)\times x . E R R = q x 2 π R 4 π ε 0 R 2 = q x 8 π 2 ε 0 R 3 \therefore E_{RR}=-\dfrac{\frac{qx}{2\pi R}}{4\pi \varepsilon_0 R^2}=-\dfrac{qx}{8\pi^2\varepsilon_0 R^3} which is the required field at centre due to remaining ring and minus sign denotes that the direction of field due to remaining ring is radially outward towards the cutout. Hence 8 + 2 + 3 = 13 \large 8+2+3=\boxed{13} .

Did the same way! Simple and elegant.

Miraj Shah - 5 years, 3 months ago

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Right .. Thanks! :-)

Rishabh Jain - 5 years, 3 months ago

Thanks for a wonderful solution. Please do replace ε { \varepsilon } with ε 0 { \varepsilon }_{ 0 } for clarity.

Swapnil Das - 5 years, 3 months ago

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Welcome and edited.. :-)

Rishabh Jain - 5 years, 3 months ago

Actually a same question is also present in "Electricity and Magnetism by Anurag Mishra". A simple yet a good question anyway

Miraj Shah - 5 years, 3 months ago

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Oh, I see. I got this one from another book. Thanks for solving.

Swapnil Das - 5 years, 3 months ago

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Which book? @Swapnil Das

Anik Mandal - 5 years, 2 months ago

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