Relationship between E and V in space

[vcsharp2003]

Homework Statement

Why would each of the following be false for $E$ and $V$ values at a point in space?
(a) if $E = 0$ then $V = 0$
(b) if $V = 0$ then $E = 0$
(c) if $V \ne 0$ then $E \ne 0$
(d) if $E \ne 0$ then $V \ne 0$

Relevant Equations

(1) $E\cos \theta = - \frac {dV} {dr}$ where $\theta$ is angle between electric field $\vec E$ and displacement $\vec {dr}$

(2) $V(\vec r) = \int_{\infty}^{\vec r} \vec E \cdot \vec {dr}$

(a) Knowing $E$, we can use equation (2) to determine $V$. However, since $\vec E$ represents the distribution of electric field in space i.e. a function of (x,y,z). For example, $\vec E = x \hat i + y \hat j + z \hat k$. Here we do not know this function so how can we know $V$ at a point?

(b) Knowing $V$ we can determine $E$ using equation (1). Again we do not know $V$ as a function of (x,y,z) so how can we determine $E$ at a point?

(c) The same issue as mentioned in (b) comes up

(d) The same issue as in (a) comes up

 

[Shreya]

Did you mean potential difference? Because potential itself is not physically significant

 

[vcsharp2003]

Did you mean potential difference? Because potential itself is not physically significant

No, I meant electric potential at a point.

 

[Shreya]

You could think of cases to disprove them. Thinking of conductors will help you.

 

[haruspex]

Knowing E, we can use equation (2) to determine V.

I think you mean "knowing $\vec E$$(not E)$ all along a line from infinity" (and assuming we are defining V to be zero at infinity).
Similarly in b, "knowing $\vec V$ over an interval".

 

[anuttarasammyak]

I observe it as a general relation of function and its derivative and think we cannot say all (a) to (d) .

 

[vcsharp2003]

I think you mean "knowing $\vec E$$(not E)$ all along a line from infinity" (and assuming we are defining V to be zero at infinity).
Similarly in b, "knowing $\vec V$ over an interval".

Yes.

Also, one could look at it as a 3 dimensional space problem. So, knowing the functions for $E$ and $V$ would tell us how the field and potential is distributed in a 3 dimensional space.. For example, let's assume a uniform electric field in space; then the electric field distribution function in 3D space would be $\vec E = e \hat i$, assuming that $e$ is magnitude of electric field and also the electric field points in the positive x- direction so there are no components along y or z axes.

Also, as an example $V = a + bx + cy^3$ where $a,b,c$ are some constants, would give us potential at any point in space i.e. it tells us the potential distribution function in space.

If we knew these functions then we could easily use equations (1) and (2) to know $E$ and $V$. It then becomes a calculus problem.

 

[vcsharp2003]

You could think of cases to disprove them. Thinking of conductors will help you.

I used your approach to answer this question instead of using rigorous calculus with equations (1) and (2).

Below is what I came up with. I considered two charges of $+1C$ separated by a distance $10 cm$ to disprove (a) and (c). I then considered the mid point of the line joining these point charges where net $E=0$ by superposition principle but $V\ne0$. This disproves (a) as well as (c).

Similarly, to disprove (b) and (d) I considered charges $+1C$ and $-1C$ separated by a distance of $10cm$. Then, I determined $E$ and $V$ at the mid point of line joining these point charges, using superposition principle.

 

[Shreya]

Below is what I came up with. I considered two charges of +1C separated by a distance 10cm to disprove (a) and (c). I then considered the mid point of the line joining these point charges where net E=0 by superposition principle but V≠0. This disproves (a) as well as (c)]

. I am an introductory physics student, so I don't know how to solve it rigorously using calculus. I will leave that to the experts

 

[Delta2]

I think the "key" point of this problem is that it asks for a point in space, that is a single point. If it was in a region of space containing infinite points (all finite regions of space contain infinite points) then things would be different. Everything can be derived from $\vec{E}=\nabla V$, having again in mind what happens in this equation when the values are as given for a single point in space.

 

[vcsharp2003]

Everything can be derived from E→=∇V, having again in mind what happens in this equation when the values are as given for a single point in space.

Is the RHS of equation you mentioned the same as potential gradient in direction of $\vec E$?

If yes, then we could say that $0=\frac {dV} {dr}$, but only for a region in space in which electric field is zero and not for a single point. We could then say that $V$ is constant for all points in this region. Therefore, $E=0$ in this region but $V \ne 0$, which disproves (a) and also (c). Is this correct?

If above is correct then the same equation does not disprove (b) and (d) since they assume $V=0$ in a region in space which always results in $\vec E=0$ since $E = -\frac {d(0)} {dr} =0$.

 

[Delta2]

Is the RHS of equation you mentioned the same as potential gradient in direction of $\vec E$?

If yes, then we could say that $0=\frac {dV} {dr}$, but only for a region in space in which electric field is zero and not for a single point. We could then say that $V$ is constant for all points in this region. Therefore, $E=0$ in this region but $V \ne 0$, which disproves (a) and also (c). Is this correct?

If above is correct then the same equation does not disprove (b) and (d) since they assume $V=0$ in a region in space which always results in $\vec E=0$ since $E = -\frac {d(0)} {dr} =0$.

Yes it is the gradient of potential that equation.

If we assume that we speaking for a region of space then indeed, (a) and (c) don't hold but (b) and (d) might hold as you say.

 

[Steve4Physics]

I suspect the question is poorly worded and should say something like this:

At a point, P, in space, each of the following statements can be true of false. Under what conditions will each statement be false? (Take potential to be zero at infinity in the usual way.)​

I would answer like this:

a) How can the statement "If E=0 then V=0" be false at P?

The statement will be false if the (vector) sum of 2 or more fields at P is zero but the (scalar) sum of the associated potentials is non-zero. E.g. P is a point midway between 2 equal point charges.

And tackle the other statements in the same way (if you can get your head around the multiple negatives in statements c) and d)!).

 

[vcsharp2003]

I suspect the question is poorly worded and should say something like this:

At a point, P, in space, each of the following statements can be true of false. Under what conditions will each statement be false? (Take potential to be zero at infinity in the usual way.)​

I would answer like this:

a) How can the statement "If E=0 then V=0" be false at P?

The statement will be false if the (vector) sum of 2 or more fields at P is zero but the (scalar) sum of the associated potentials is non-zero. E.g. P is a point midway between 2 equal point charges.

And tackle the other statements in the same way (if you can get your head around the multiple negatives in statements c) and d)!).

I already came up with a similar solution as explained in an earlier post#8, where I assumed a pair of point charges of +1C for disproving (a) and (c), and a pair of point charges +1C, -1C for disproving (b) and (d). Separation of charges was 10 cm in both scenarios.

 

[Shreya]

I already came up with a similar solution as explained in an earlier post#8, where I assumed a pair of point charges of +1C for disproving (a) and (c)

You could also find more scenarios for the questions. I think that you cannot disprove a statement like this rigorously. Since the statement, say a, only says that if E=0, V needn't be zero. You will have to satisfy yourself with counter examples. These questions are meant to help you remove any misconceptions regarding the topic.

Please do correct me if I am wrong

 

[vcsharp2003]

You could also find more scenarios for the questions. I think that you cannot disprove a statement like this rigorously. Since the statement, say a, only says that if E=0, V needn't be zero. You will have to satisfy yourself with counter examples. These questions are meant to help you remove any misconceptions regarding the topic.

Please do correct me if I am wrong

Another example that disproves statements (a) and (d) is as below.

Inside a charged solid spherical conductor with radius $R$ and total charge $Q$, $E =0$ but $V = \frac {kQ} {R} \ne 0$.

 

[Steve4Physics]

I already came up with a similar solution as explained in an earlier post#8, where I assumed a pair of point charges of +1C for disproving (a) and (c), and a pair of point charges +1C, -1C for disproving (b) and (d). Separation of charges was 10 cm in both scenarios.

My apologies – I could have referenced your Post #8 as the example. Have a thumbs-up for Post #8!

But it’s worth noting that the original question asks ‘why?’, so specific examples alone may not be sufficient.

 

[Office_Shredder]

I think a counterexample is a pretty good description of why. I assume the point is you don't get full credit just for guessing correctly.

 

[vcsharp2003]

You could also find more scenarios for the questions. I think that you cannot disprove a statement like this rigorously. Since the statement, say a, only says that if E=0, V needn't be zero. You will have to satisfy yourself with counter examples. These questions are meant to help you remove any misconceptions regarding the topic.

Please do correct me if I am wrong

We could also use Calculus to disprove the statements. We know that the following relationship always holds between electric field and potential at all points in space.$$E_{x} \hat i +E_{y} \hat j + E_{z} \hat k = - \frac {\partial V} {\partial x} \hat i - \frac {\partial V} {\partial y} \hat j - \frac {\partial V} {\partial z} \hat k$$Let's consider the following distribution functions which satisfy the above relationship.$$E(x,y,z) = y \hat i + x \hat j$$ $$V(x,y,z) = -xy + 10$$Now, we can take each statement one at a time and verify that its false.

Consider statement (a) where we start with $E=0$. In our above example, $E(0,0,2) = 0 \hat i + 0 \hat j = 0$ , but $V(0,0,2) = (0)(0) + 10 = 10$. Thus, we came up with distribution functions such that when $E=0$ then $V \ne 0$, so statement (a) is not always true.

Now, consider statement (b) where we start with $V=0$. In our above example, $V(5,2,4) = -(5)(2) + 10 = 0$, but $E(5,2,4) = 2 \hat i + 5 \hat j \neq 0$. Thus, we came up with distribution functions such that when $V=0$ then $E \ne 0$, so statement (b) is not always true.

Similarly, we can take each of the remaining statements.

But I feel that the counter examples I gave in posts# 8 and 16 are the easiest ways to disprove these statements as they are simple and straightforward.

 

[kuruman]

But I feel that the counter examples I gave in posts# 8 and 16 are the easiest ways to disprove these statements as they are simple and straightforward.

You have to be careful with counterexamples and stick to the more rigorous ways of thinking about this which, as has been already been pointed out, relies on the idea that if a function is zero its derivative is not necessarily zero and vice-versa. In #16 you assert that

Another example that disproves statements (a) and (d) is as below.

Inside a charged solid spherical conductor with radius $R$ and total charge $Q$, $E =0$ but $V = \frac {kQ} {R} \ne 0$.

Remember that the zero reference of electric potential is only conventionally taken at infinity; the zero reference can be taken anywhere one pleases. If it pleases one to take the zero of potential on the conductor, your counter argument is invalid as justification. The truth or falsity of the four choices cannot depend on where one chooses the zero of electric potential.

 

[vcsharp2003]

Remember that the zero reference of electric potential is only conventionally taken at infinity; the zero reference can be taken anywhere one pleases.

Why do Physics textbooks assume zero potential at infinity rather than at some other point that's close to the charges being considered? This concept of taking any reference point can be confusing. I thought potential at infinity makes sense since at such large distances between charges there is almost zero electrostatic force between the charges in question.

 

[kuruman]

Yes, it is less confusing to take electrostatic potential to be zero at infinity, but that is a matter of convenience not because the force between two charges is zero when they are infinitely far apart. When you shoot a projectile from the surface of the Earth, it is more convenient (and much less confusing) to choose the zero of energy at the starting point of the projectile rather than at infinity where the force between the projectile and the Earth is zero.

 

[vcsharp2003]

The truth or falsity of the four choices cannot depend on where one chooses the zero of electric potential.

Doesn't above statement mean that no matter what approach we take i.e. counter-example or a rigorous approach, we can never prove or disprove any of the statements? Even the rigorous method would need one to select a point in space for zero potential.

 

[kuruman]

Doesn't above statement mean that no matter what approach we take i.e. counter-example or a rigorous approach, we can never prove or disprove any of the statements? Even the rigorous method would need one to select a point in space for zero potential.

I think the point of this question is that one cannot use the value of a function and its derivative at a single point in space to conclude which could or could not be zero. One has to look how the function changes with respect to an appropriate variable in order to ascertain what must and must not be zero. Where one picks the origin so that the function is zero at a desired point is irrelevant to how the function changes at any point.

 

[vcsharp2003]

One has to look how the function changes with respect to an appropriate variable in order to ascertain what must and must not be zero.

You meant how potential changes with position in space or the electric field changes with position?

 

[Shreya]

You meant how potential changes with position in space or the electric field changes with position?

I think he is referring to any general function and emphasizing that the choice of Origin doesn't influence the change (the derivative) of a function at a point. And this holds for not just E and V but any function & derivative function.
Please do correct me if I am wrong

 

[vcsharp2003]

Origin doesn't influence the change (the derivative) of a function at a point.

That's easy to see since the derivative is just the slope at a point when function is graphed against the variable and surely doesn't depend on what origin we have. But, in this question it seems to apply to potential since derivative of a potential function relates to E, but derivative of E function doesn't relate to potential.

Honestly, this rigorous way of answering my original question is quite confusing. More confusing is that none of the counter examples are valid since we can change the zero reference point. When a question says potential, it automatically means it's relative to some value.

 

[Steve4Physics]

Honestly, this rigorous way of answering my original question is quite confusing. More confusing is that none of the counter examples are valid since we can change the zero reference point. When a question says potential, it automatically means it's relative to some value.

You can ‘t (IMHO) rigorously answer such an ill-posed question. You will go round in circles (or ellipses in the case of this erudite forum).

One approach is to state whatever assumptions make a solution feasible and then proceed on that basis, as others have already done. (E.g. see my Post #13.)

Different interpretations of the question will inevitabl;y lead to different answers.