# Should point at which E field = 0 coincide with V = 0?

A problem I solved asks this:

There is a point charge of 3q at the origin and another charge of -2q at x = 5. At what value of x relative to the origin is the electric field equal to zero?

So:

E = E1 + E2
0 = E1 + E2
E1 = -E2
kQ1/x2 = -kQ2/(5-x)2

Solving for x, the point at which these fields cancel out is x ≈ 2.75.

At the start of this problem I had mistakenly solved for the point at which the electric potential equals zero:

V = V1 + V2
0 = V1 + V2
V1 = -V2
kQ1/x = -kQ2/(5-x)

Solving for x yields a clean x = 3.00.

Are my calculations correct, or should these points coincide? At a point in which the total electric field equals zero, shouldn't the total voltage be zero as well?

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## Answers and Replies

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BvU
Homework Helper
2019 Award
hi,
if you look at the relationship between potential and electric field strength (or at the dimensions), you see that E = 0 if ΔV = 0 for a (small) change in position...

BvU
Homework Helper
2019 Award
One more thing is that V is only established up to a constant. We usually choose that to have V = 0 at infinity, but that is just a convention.

hilbert2
Gold Member
There's two points on the x-axis where the electric field is zero (if not including the "point" at $x\rightarrow\pm\infty$), and neither of them is between the charges when $Q_1$ and $Q_2$ are of different sign as in this problem.

Chandra Prayaga
Ask your self the following question in 1-dimension. If a function of x is zero at a certain value of x, is its slope equal to zero at the same value of x?

haruspex
Homework Helper
Gold Member
Ask your self the following question in 1-dimension. If a function of x is zero at a certain value of x, is its slope equal to zero at the same value of x?
Just to clarify, the OP's question is why isn't V zero where E is zero, so the directly analogous question in 1D is the reverse of the above: why isn't the value of the function zero where its slope is zero?

Further to the above replies, there is nothing special about a point where V is zero. Potentials are always relative. You can define any point to be at V=0; the potential at any other point is relative to that.
In many electrostatics questions it is common to define the potential at infinity to be zero, but it is only a convention.
As against that, E=0 does have significance.

Ask your self the following question in 1-dimension. If a function of x is zero at a certain value of x, is its slope equal to zero at the same value of x?
That really flicked a switch in my head. Thanks for that!