Two point charges of equal magnitude Q are held a distance d...

Two point charges of equal magnitude $Q$ are held a distance $d$ apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?). (b) Repeat part (a) for two point charges having opposite signs.

Two point charges of equal magnitude $Q$ are held a distance $d$ apart. Consider only points on the line passing through both charges. (a) If the two charges have the same sign, find the location
of all points (if there are any) at which (i) the potential (relative to infinity) is zero (is the electric field zero at these points?), and (ii) the electric field is zero (is the potential zero at these points?).
(b) Repeat part (a) for two point charges having opposite signs.

Transcript

00:01 Here's a nice little exercise to work with point charges and also demonstrate the difference between potential and electric field.

00:10 So we have two point charges to start off with separated by distance d along the x -axis.

00:19 And we want to find out where the potential total is zero.

00:25 And to do that, we would have to add potentials from both charges.

00:35 So we'll add those together using the point charge formula, which has the charge in the numerator with the constant k and the distance from each charge.

00:53 So we have to ask, the distances are, of course, always positive.

01:00 That appear in the denominator.

01:03 And let's just show what that would be.

01:05 So we're only considering points along the line, joining the two.

01:11 So d1 would be, say, to the left of charge one, and d2 would be taking care of that.

01:26 Let's see, d plus d1 in this case, but it could be d minus d1, if it was in between.

01:35 It really doesn't matter, but we'll see why a little bit.

01:40 The numerator in this case for both terms is always positive, and the denominator involving distances is always positive.

01:53 And so there's no way for those two potentials to add up to zero unless the observation point x is at plus or minus infinity.

02:13 And that's kind of disappointing because we want to find points that are not at an infinity.

02:24 Okay, so the potential cannot be zero in a finite distance.

02:37 Let's take a look at the electric field.

02:40 So electric field can be added from both.

02:45 And we usually have the charge with an absolute value.

02:53 Again, there's a distance squared in the denominator.

02:57 And then there's usually some little unit vector on the front of that to demonstrate that the electric field is a vector, of course, and points in the line from the source to the observation point.

03:19 So let's take a look at some of the observation points.

03:24 In between the two charges, what will happen is, let's call these two charges 1 and 2.

03:32 So we can separate them out.

03:35 The electric field from charge 1 will point to the right and from up charge 2, it will point to the left.

03:48 And so if those are equal in magnitude, they will cancel.

03:54 So that tells us that the electric field is zero if d1 is equal to d2...

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