Calculating Work Done per Unit Charge in an Electric Field

[brake4country]

Homework Statement

A charged particle is moved from a great distance to a distance d from a point charge. At distance d, the electric field has a strength E and a potential V. Which of the following represents the work done per unit charge q?

Homework Equations

F = Eq; V = Ed, E = kq/r^2; U = Eqd, U = kq1q1//r

The Attempt at a Solution

In a previous thread, it was learned that the potential energy of a charged particle is denoted as U = Vq, where q is the charge being affected by the electric field. Now, a slightly different scenario is presented. The same particle is moved from an infinite distance to a distance d to a point source charge. The amount of work done here is the W = Eqd, a derivative of F=ma.

The main difference between the problem before and the problem here is the "distance". If we know the distance the charge was brought from far away to nearer to the point charge, then we can calculate the work done on the particle. This is the formula U =Eqd.

Is my thought processes correct here? Thanks in advance!

 

[haruspex]

W = Eqd

That's only true for a constant field E acting in the same direction as the distance moved, d.
In general, $W=\int\vec E.\vec {ds}$, where $\vec{ds}$ is the distance element vector.
It isn't entirely clear what the question is asking for . It says 'per unit charge', but there are two charges here.
What dimensionality would 'work per unit charge' have? Which, if any, of the offered answers has that dimensionality?

 

[brake4country]

I see how the question is ambiguous. To answer your question, the dimensionality for work per unit charge is W/C = Fd/C = Eqd/C. The answer choices are (A) V, (B) Eqd, (C) Vq, and (D) Ed/q. A is wrong because the dimensionality for V is volt or J/C. C is wrong because Vq is the potential energy charge, which we are not asked for, and (D) is Force per charge, which is wrong because Force is not work although it is related via distance d. The best answer is B.

 

[haruspex]

I see how the question is ambiguous. To answer your question, the dimensionality for work per unit charge is W/C = Fd/C = Eqd/C. The answer choices are (A) V, (B) Eqd, (C) Vq, and (D) Ed/q. A is wrong because the dimensionality for V is volt or J/C. C is wrong because Vq is the potential energy charge, which we are not asked for, and (D) is Force per charge, which is wrong because Force is not work although it is related via distance d. The best answer is B.

I agree that's the only one which makes sense dimensionally, but it is clearly wrong. What would it give for the work done when d is zero?
So I tend to think the question has been badly worded - they want the work, not the work per unit charge. That suspicion is reinforced by the mention of q at the end of the sentence. If they wanted the work per unit charge, there would no relevance in mentioning q.
This leads me to the last option, but there seems to be at least one typo in it. Please check that you have posted it correctly.

 

[BvU]

First you want to fix the typo in equation 4. What is q1 anyway ?

You haven't mentioned the choices in your original post, even though they are essential. That way the question is not ambiguous at all.

Your dimensional reasoning is wrong. Work has the dimension kg m²/s² , in short: Joule
(C) has the dimension of Work (=the dimension of potential energy change)
The dimension of (D) is not Force per unit charge. Think again.

 

[BvU]

I see that even Haru was lured into believing your equations were the choices.

 

[haruspex]

I see that even Haru was lured into believing your equations were the choices.

Ah! Yes, I didn't notice the four later formulae were different from those in the OP. Thanks BvU.
Your list of dimensional representations finishes with Eqd/C, but q and d there are actual variables, not dimensionalities. Fix that and you should get some cancellation. What remains can be represented in a single concept.

 

[brake4country]

I agree that's the only one which makes sense dimensionally, but it is clearly wrong. What would it give for the work done when d is zero?
So I tend to think the question has been badly worded - they want the work, not the work per unit charge. That suspicion is reinforced by the mention of q at the end of the sentence. If they wanted the work per unit charge, there would no relevance in mentioning q.
This leads me to the last option, but there seems to be at least one typo in it. Please check that you have posted it correctly.

Yes, I agree that it was poorly worded but Exam Krackers is known for that. It should say "per unit charge." Thank you so much!

 

[BvU]

In post #5 I argue that the question with the choices as shown in your post #3 is not ambiguous at all. It is also properly worded.
Is it now clear to you which answer is correct, and why ?

 

[brake4country]

The answer choices are in post #3. Yes, I understand now. Thank you everyone.