If \(\vec{E}\) is given by\(\vec{E} = y \hat{i} +x \hat{j}\) , then find the the potential \(V_{(x,y)}\) if \(V_{∞}\) is 10 \(V\).

I did

\[ \int_{10}^{V_{(x,y)}} dV= -\int_{∞}^{(x,y)} (y \hat{i} +x \hat{j}).(dx \hat{i} +dy \hat{j})\]

Substitute \(z=xy\)

\[V_{(x,y)} - 10 = [z]^{xy} _ ∞\]

\[\color{red}{V_{(x,y)} - 10 = xy - ∞}\]

Why that happened

[Edit] - Issue resolved , thanks soumo for help.

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## Comments

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TopNewestI want you to spot a fallacy (if it's there) in the following :

\[ { V }_{ \left( A \right) }-{ V }_{ \left( B \right) }=-\int _{ B }^{ A }{ E.dr } \\ { V }_{ \left( A \right) }-{ V }_{ \left( \infty \right) }=-\int _{ \infty }^{ A }{ E.dr } \\ { V }_{ \left( A \right) }=-E.A+E.\infty \]

I have assumed \(\displaystyle { V }_{ \infty }=0\) and \(E\) to be constant.

So, what is wrong? Is even anything wrong?

We know that \(\displaystyle { V }_{ \left( A \right) }-{ V }_{ \left( B \right) }=-\int _{ B }^{ A }{ E.dr } \). To find the electric potential at \(A\) we choose \(B\) to be a reference point. It is very nice if we choose \(B\) such that the electric potential at \(B\) is zero. According to your \(E\) (electric field), potential is zero at \(x=0,y=0\) i.e., at the origin.

So do we need to find \(\displaystyle { V }_{ \left( A \right) }-{ V }_{ \left( \infty \right) }\)? or \(\displaystyle { V }_{ \left( A \right) }-{ V }_{ \left( 0 \right) }\) i.e. \(\displaystyle { V }_{ \left( A \right) }\)

Only after making some assumptions, I came across the following approach:

If \(\displaystyle { V }_{ \left( A \right) }-{ V }_{ \left( B \right) }=-\int _{ B }^{ A }{ E.dr } \)

Then \(\displaystyle { V }_{ \left( A \right) }-{ V }_{ \left( 0 \right) }=-\int _{ 0 }^{ A }{ E.dr } \) gives the potential at \(A\) w.r.t origin \(0,0\).

On the same basis \({ V }_{ \infty }-{ V }_{ \left( 0 \right) }=-\int _{ 0 }^{ A }{ E.dr } \) gives the potential at infinity w.r.t to origin.

And \(\displaystyle \left( { V }_{ \left( A \right) }-{ V }_{ \left( 0 \right) } \right) -\left( { V }_{ \infty }-{ V }_{ \left( 0 \right) } \right) \) gives potential at \(A\) w.r.t infinity.

Hope this helps you to find the answer(s) even if this isn't the answer.

:)

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The problem is that, if I consider any point in x-y plane I get answer of electric potential as -infinity .I am unable to figure out what's the problem.

Why is relation between x and y required?

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The ultimate aim is to find the potential at a point w.r.t another point where we can reasonably take the potential to be zero. This reference point need not necessarily be infinity.

I wanted you to spot the fallacy in second point of my previous comment. That way we always get potential at any point to be infinity. That approach is wrong.

Did you try this \(\displaystyle \left( { V }_{ \left( A \right) }-{ V }_{ \left( 0 \right) } \right) -\left( { V }_{ \infty }-{ V }_{ \left( 0 \right) } \right) \)?

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Here Electric field at infity is infinite , so you can't take potential reference as 10 volt at that place (region) . Since E.F here has trajectory of hyperbola , which is unbound at very very very large distance from origin , So potential at infinite being 10 volt is meaningless .

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I didn't got why or how it is meaningless

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@Soumo Mukherjee @Raghav Vaidyanathan @Nishant Rai @Ninad Akolekar

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