Two oppositely charged identical insulating spheres, each 50.0 cm in diameter and carrying a uni

Two oppositely charged identical insulating spheres, each...

Two oppositely charged identical insulating spheres, each 50.0 $\mathrm{cm}$ in diameter and carrying a uniform charge of magnitude $175 \mu \mathrm{C},$ are placed 1.00 $\mathrm{m}$ apart center to center Fig. 18.53 ). (a) If a voltmeter is connected between the nearest points $(a$ and $b)$ on their surfaces, what will it read? (b) Which point, $a$ or $b,$ is at the higher potential? How can you know this without any calculations?

Key Concepts

Qualitative Analysis of Potential Differences

Understanding the sign of the charge allows one to determine the direction of the electric field and how the potential varies spatially. For instance, a positive charge creates a potential that decreases with distance, while a negative charge creates a potential that is more negative closer to the charge. Such qualitative reasoning enables one to decide which point is at a higher or lower potential without detailed calculations.

Electric Potential

Electric potential is the energy per unit charge at a point in an electric field, typically measured in volts. It represents the work done in moving a small test charge from a reference point (usually infinity) to that point. In problems involving charged objects, the potential due to each object can be calculated and then combined using appropriate methods.

Superposition Principle for Potentials

The superposition principle states that the total electric potential at a point caused by multiple sources is the sum of the individual potentials produced by each source independently. This principle holds because electric potential is a scalar quantity, making the addition straightforward without regard to direction.

Transcript

00:01 In this question, we have two spheres that is equal but with negative charges.

00:07 So i will call the spheres radius r, and then i will say this has positive q and this has negative q.

00:15 And they are separated by a distance.

00:20 Now what happens is i will call this point a, this point b, and i have a voltmeter that connects a and b.

00:28 And i want to know what numbers this word media will give me.

00:32 So one really, really important thing in this question is that these spheres are insulating spheres.

00:40 What does that mean is that for a conducting sphere, we would expect that the entire sphere will be at the same potential.

00:47 But that's not true for insulating sphere.

00:50 For an insulating sphere, say the potential here will be very different from the potential here, because charge do not flow on an insulating sphere.

00:59 What that also makes is that in order to find the potential of point a, you should not, it is technically on a sphere, but you should also just treat it as if it's, treat it's potential as if just a simple superposition of this point charge and this point charge.

01:23 Which is not true for conducting sphere.

01:25 For a conducting sphere, the entire sphere will actually share a same potential due to the other charge.

01:33 So that was more complicated, but here, because it's an insulating sphere, you just brute falsely finds the potential due to the positive charge, and you add them together.

01:45 So we'll give you the potential at point a.

01:47 And you do the same for the potential at point b, and you subtract them.

01:50 So that gives you the potential difference, which is what the vote meter will tell you.

01:56 Of talking, let's jump into actually doing this question.

02:00 So for the potential at a due to this positive charge is kq over r, because this is r.

02:08 So this point, the potential of a due to this is kq over r.

02:17 And the potential at point a due to the negative charge is this whole distance, because this thing is d.

02:27 This is d.

02:30 So a to the center of the sphere is d minus r.

02:35 So the potential at a due to the negative charge is negative kq over d minus r.

02:45 And same with vb...

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