HW 03 - Electrostatic Potential

1 1 1 Equation Chapter 1 Section 1 HW 03 - electrostatic potential : chapter 24, problems 8, 17, 25, 33, 39, 51, 59, 74 therein.

•• Chapter 24, problem 8: A graph of the x-component of the electric field as a function of x in a region of space is shown in the Figure. The scale

of the vertical axis is set by . The y and z components of the electric field are zero in this region. If the electric potential at the origin is

, (a) what is the electric potential

at , (b) what is the greatest positive value of the electric potential for points on the x axis for which , and (c) for what value of x is the electric potential zero?

(a) The electric field is , where and is the unknown function of that is the

solution to the problem. In the region , the linearly-increasingly-negative electric field is given by the

linear function (this function is the solution to the question “what linear function passes

through the origin and through the point ?”). This is an “ordinary” differential

equation with the “boundary condition” to coin some lingo from differential equations class1,

212\* MERGEFORMAT (.)

Carrying out the integral on the right hand side (“RHS”) of 12 and solving for ,


1 That sounds like an advanced topic, but I’m only mentioning some lingo from it (rather than gory computational techniques). You heard it here first, so it won’t be so scary when you hear it later…

(b) The problem is asking for the global maximum of the potential given a piecewise-defined electric field, so we must find the potential in all regions. The functional form of the electric field is2,

Then, the potential in all 3 regions is similarly piecewise-defined3,

Notice that the integration in the 2nd column gives you the potential starting from due to the bounds of

integration starting at , and the integration in the 3rd column gives you the potential starting from

due to the bounds of integration starting at . Hence the term “boundary” condition.

We now shall carry out the integration. To make the integration cleaner4, let’s effect the change of integration-

variables ,


From 14 we can calculate5 the boundary condition , which is needed in the expression for the potential in the 3rd region,


We can plot both the electric field and the potential it follows from vs. the dimensionless . In tabular form, these objects appear as, 2 In the 2nd column, the red “3” was incorrectly a “2” before! It was a mistake I made while doing the problem, and I only noticed the mistake by doing this problem out in gory detail as I have. I encourage you to do the same when tackling problems.

3 In the 2nd and 3rd columns of the above table, is what people call a “dummy (integration) variable”: it is a placeholder-variable naturally occurring in the integration process, and people encounter it so much that it has its own name. You’ll hear people talk about “dummy variables” all the time.4 You may be squeamish about unnecessarily-effecting substitution-integration, but “clean” means less chance of making a mistake!5 The red “ ” in the last step is another place where I made a mistake! It cost me a lot of time while preparing this solution set, and it could have cost me more if I didn’t explore this problem deeply and record in writing my exploration.

Interval

Electric fieldpotential

It is obvious that we should plot the dimensionless electric field and the dimensionless potential

, and we hence notice that the plot is sensitive to the ratio6 ,

0.5 1.0 1.5 2.0 2.5 3.0 xx1

1

1

2

3

4V V0,EES Potential and Field vs. x

VxV0ExES


(c) With the plot 16, one can numerically see where the potential passes through zero, but we can solve the problem analytically just to be sure. Let be the coordinate where the potential is zero. We also see that is

in region-3, so . Solving this expression for ,


Afterword: I made lots of mistakes in preparing the above solution (Obviously, those mistakes are corrected, and I do not show them to you because they’d cause you all sorts of mischief and trouble when you’re trying to learn), and I was able to quickly correct them by laying out and carefully recording all my works and steps!

6 Hence, the relative shape of the plot depends on only one of the problem gives! A 2-dimensional plot is a way to illustrate a salient feature of a problem, and plotting dimensionless quantities automatically removes up to 2 of the “givens” in the problem! For instance,

our problem-statement contains 3 givens ( , , and ), but by plotting vs.

and vs. , we see that the plot is sensitive only to (a dimension-ful quantity of inverse-length).

Chapter 24, problem 17: In the Figure, what is the net electric potential at point due to the four

particles if at infinity, , and

?

In contrast to the above, this is a rather boring question to answer, for we just write,


Afterword: Notice a few things about our calculation: (1) The electric field can be gotten from the potential by

the operation . Although aren’t floating around in our expression 18, this is possible, but we would require a coordinate system. That brings us to (2) Even so a potential implies an electric field, calculating the potential does not necessarily7 make reference to a coordinate system (unlike the electric field it implies)!

Chapter 24, problem 25: A plastic rod has been bent into a circle of radius . It has a charge

uniformly distributed along one-quarter of its circumference and a charge

uniformly distributed along the rest of the circumference (see the Figure). With at infinity (i.e., ), what is the electric potential at (a) the center C of the circle and (b) point P, on the central axis of the circle

at distance from the center?

(a) Let be the origin of a cylindrical coordinate system. The potential at is due to a charge-density

in the circle-arc of length plus that of a charge density in the

circle-arc of length . An element of charge (where is a polar-

coordinate-angle) is a distance away, where is piecewise-defined over , and so,

7 One does make reference to a coordinate system in calculating the potential if they do wish to have floating around in their (simplest exact form) expression, because then one would apply the operation described above…to get a field.


Plugging in numbers, we have,


(b) For a point a distance above , we simply repeat the same steps in 19, except with the replacement

effected in the denominator where it appeared in 19,


Plugging in numbers, 111 yields,


••• Chapter 24, problem 33: The thin plastic rod

shown in the Figure has length and

a nonuniform linear charge density ,

where .With at infinity, find

the electric potential at point on the axis, at

distance from one end.

Even so this is a 3-dot problem, it is physically trivial (only the calculus is difficult) if we follow the same procedure that we did for any similar 1-dot problems we may have undertaken,


Math interlude – for your interest only 8 : We then integrate by parts by (1) calculating the differential

using the product rule and solving for the resulting , as,


Then, (2) calculating the differential and solving for the resulting in which ,


End of mathematical interlude: Combining 114 and 115 with 113, and calculating the dimensionless9 and , we have,

16116\*MERGEFORMAT (.)

Re-introducing and , we have,


•• chapter 24, problem 39: An electron is placed in an xy-plane where the electric potential depends on x and y as shown in the Figure (the potential does not depend on z). The scale of the vertical axis is set

by . In unit-vector notation, what is the electric force on the electron?

8 Meaning I will not quiz you on the purely-mathematical machinations given here. I will just give you the antiderivative. This is because this is a physics class, not a calculus class! But it’s good, also, to gain a sense of which problem-solving steps are mathematical vs. those which are physical. (Often, the latter are harder!).9 Obviously, scales out the computationally-intensive machinations of calculus we are effecting from 114 and 115.

Let the scales of the horizontal and vertical axes be , as indicated. Then,


Chapter 24, problem 51: In the rectangle of the Figure, the sides have lengths and

widths , , and

. With at infinity, what is the electric potential at (a) corner A and (b) corner B? (c) How much work is required to move a charge

from B to A along a diagonal of the rectangle? (d) Does this work increase or decrease

the electric potential energy of the three charge system? Is more, less, or the same work required if

is moved along a path that is (e) inside the rectangle but not on a diagonal and (f ) outside the rectangle?

(a) and (b) Let , where we obviously have and where . Now we won’t have any ’s with subscripts running around—only Greek letters and terms of . We further note that if we had (which we do not) the answers to parts (a) and (b) would be the same, and we can use this as a

check against any mistakes. This being said, let us proceed to calculate ,


(c) and (d) The work to move a charge from B to A is easily calculated as,


The work 120 calculated is defined to be that which an external agent does. When an external agent does positive work upon a system, the energy of that system increases10.

10 In our case, potential energy is stored in the charge system, which may be realized as acceleration for the charges being massive. Other examples: friction does positive work upon a sliding object, and the “work input” is realized as thermal energy.

(e) and (f) The electrostatic force is conservative, so the work is the same no

matter which path is used. Proof (pure math – for your own interest): A conservative force field is a

vector field whose vector curl is zero. The curl in this special case is given by

, wherein we have and , and so11,


Chapter 24, problem 59: In the Figure, a charged particle (either an electron or a proton) is moving

rightward between two parallel charged plates separated by distance . The plate potentials are

and . The particle is slowing from an initial speed of at the left plate.

(a) Is the particle an electron or a proton? (b) What is its speed just as it reaches plate 2?

We do part (b) first where we calculate the final velocity , and note that because it is said that the

particle is slowing. We furthermore note that the mass is that of either an electron or a

proton . Hence,


In 122, for , we must have ; since , we have

, and thus the particle is a proton; accordingly, the final velocity 122 is,


We do not need the distance between the plates, because of the path-independence of the conservative electrostatic potential, and because “what’s the final velocity” is a question about a final state,

11 In the last step, we used the fact that derivatives commute: . The is called the gradient operation, and the curl of a gradient always vanishes. You won’t be quizzed on this proof, but my mentioning it may make Calc III easier.

rather than what happens in between. We would need the distance if we wanted to calculate quantity that happens in between the two plates, such as the acceleration of the proton during its trip.

Chapter 24, problem 74: Three particles, charge , , and

, are positioned at the vertices of an isosceles triangle as shown in the Figure. If

and , how much work must an external agent do to exchange the positions of (a)

and and, instead, (b) and ?

The answer to part-a is , and the answer to part-b is , where is the total energy

of the charge-distribution, is the total-energy of the charge-distribution with interchange, and is the total-energy of the charge-distribution with interchange. These three expressions are,


From 124, we immediately see that , which is the answer to (b) in simplest exact form12.

For part-a, letting13 , , and (i.e., ,

25125\*MERGEFORMAT (.)

Plugging in numbers to 125, we get,


12 That is, the result for part-(b) is insensitive to the choices of the numbers at the outset of the problem (what it is sensitive to is the chosen geometry of an isosceles triangle; the zero-answer here is a consequence of its symmetry).13 I’m just letting Greek letters be dimensionless multiples that tell the relative values of one quantity to another. For instance, the fact

that is “triple the charge of ” means ; similarly with the I introduce for the lengths .

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HW 03 - Electrostatic Potential