Infinite Potential

Electricity and Magnetism Level 2

An infinite number of concentric rings carry a charge Q Q each alternatively positive and negative. Their radii are a , 2 a , 4 a , 8 a − − − − m e t e r s a, 2a, 4a, 8a ---- meters in geometric progression as shown. where a = 1 m e t e r a=1 meter . The potential at the center of the rings will be?

Q 6 π ϵ \frac{Q}{6\pi \epsilon} Q 4 π ϵ \frac{Q}{4\pi \epsilon} 0 0 Q 12 π ϵ \frac{Q}{12\pi \epsilon} Q 8 π ϵ \frac{Q}{8\pi \epsilon}

Abhay Tiwari by Abhay Tiwari , 28, India

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1 solution

Potential at the centre is given by : \text{Potential at the centre is given by : }

V = 1 4 π ϵ ( Q a − Q 2 a + Q 4 a − ⋯ ) \displaystyle V=\frac{1}{4\pi\epsilon}(\frac{Q}{a}-\frac{Q}{2a}+\frac{Q}{4a} - \cdots)

⟹ V = Q 4 π ϵ a ( 1 − 1 2 + 1 4 − ⋯ ⏟ Infinite GP ) \implies\displaystyle V = \frac{Q}{4\pi\epsilon a}(\color{#3D99F6}{\underbrace{1-\frac{1}{2}+\frac{1}{4} - \cdots}_{\text{Infinite GP}}})

⟹ V = Q 4 π ϵ a ( 1 1 + 1 2 ) = Q 4 π ϵ a 3 2 = Q 6 π ϵ \implies\displaystyle V = \frac{Q}{4\pi\epsilon a} (\frac{1}{1+\frac{1}{2}}) = \frac{Q}{4\pi\epsilon a}\frac{3}{2} = \boxed{\frac{Q}{6\pi\epsilon }}

7 Helpful 1 Interesting 1 Brilliant 0 Confused

How did that 6 came in the last step cause the value of a is 1 so how does that happened?

Veer Pratap Singh Rathore - 1 year, 10 months ago

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