Two tiny objects with equal charge of 7.00 μC are placed at two corners of a square with sides o

Two tiny objects with equal charge of 7.00 μC are placed at...

Key Concepts

Coulomb's Law

This fundamental law describes the interaction between two point charges. It states that the magnitude of the electric force between charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is the basis for calculating the electric field produced by a single charge in space.

Electric Field of a Point Charge

The electric field at any given point in space due to a point charge is defined as the force per unit positive test charge placed at that point. It is given by Coulomb's law and has both magnitude and direction, pointing away from positive charges and toward negative charges. This concept is essential when considering how individual charges influence the space around them.

Principle of Superposition

This principle states that the total electric field created by multiple charges is the vector sum of the electric fields produced by each individual charge. In problems involving several charges, like the one described, it allows for the calculation of the net electric field by considering the contribution of each charge separately and then summing them as vectors.

Vector Addition

Since electric fields are vector quantities, their combined effect is determined using vector addition. This involves summing the magnitudes and directions of each individual field contribution to determine the net field at a specific point. Understanding vector addition is crucial in analyzing scenarios where the directions of the electric fields differ due to different charge locations.

Symmetry in Electrostatics

Symmetry plays a vital role in simplifying electrical problems by reducing the complexity of the vector addition process. When charges are arranged in a symmetric configuration, certain components of the electric fields may cancel out, making it easier to determine the net field. Recognizing symmetry can help identify special positions or directions where the net electric field may be zero.

Transcript

00:02 All right, in this video, the concept we're going to look at is the electric field of a point charge.

00:09 Okay, so we're going to have three different point charges.

00:12 All are going to have the same charge.

00:15 And we're going to place a third charge so that we can get a zero electric field at a spot.

00:22 So we're going to have a square.

00:29 And at these bottom two corners, we're going to have charge q.

00:35 And at this corner here, point a, we want to add a third charge to make no field at that corner.

00:47 All right.

00:49 So each q is 7 .00 microcouliams.

00:59 And a microculem is a 10 to the minus 6.

01:04 So let's just convert that now.

01:08 And the square, each side has a length of.

01:13 Of 0 .3 0 .30 meters.

01:21 Okay, so both of the charges are positive, right? the third charge is going to be positive, too.

01:29 And so the q in the lower left is going to make a field straight up and away from a.

01:38 Okay, so let's draw that over here.

01:43 Let's just call that e1, and this is corner one, corner two.

01:52 Now, the field by the charge at corner 2 is going to be right along the diagonal connecting a, corner a, and q2.

02:08 So it's going to be going off like this.

02:11 We'll call that e2.

02:15 It will have components, and let's just draw this in e2x, which is in the negative direction, and e2y, which is in the positive direction.

02:35 Okay, so we need to add another charge to make field zero there.

02:40 So it has to be at least over as much as e2x and down by the amount e2y plus e1.

02:52 So it has to balance the other two electric fields.

02:59 So we'll call that e3.

03:01 This here is e3x and e3y, and it goes at some angle theta below the x -axis.

03:15 All right.

03:18 Now, since both of the sides, the side between all four sides are the same distance, that means this angle here is 45 degrees.

03:29 Same with this one here, 45 degrees.

03:34 All right.

03:35 And so let's write down the e2x plus the e3x are going to add up to zero.

03:47 And so that means e3x is the negative of e2x.

03:54 Okay.

03:56 And then we also have that e1, which is all in the y direction, plus e2, 2y plus e3y, i'll have to add up to 0.

04:16 So e3y is minus e1 minus e2y.

04:23 So we need to find the components.

04:27 Whoops, wrong one.

04:34 We need to find these components.

04:44 All right, let's start with the y's, i guess.

04:46 Okay, so we'll start over here.

04:49 Okay, so for the q at corner one, we have the electric field, e1 is k times q, k times q over r squared.

05:09 The r is the distance between the charge and where you want to know the electric field.

05:15 So that distance is d.

05:18 Okay, so that we can find e1.

05:22 Is k.

05:23 K is 8 .99 times 10 to the 9 newton meters squared per kulams squared.

05:32 The q we said was 7 microculems, so 7 .00 times 10 to the minus 6 couloms.

05:41 And then the distance is 0 .30 meters and we need to square that.

05:50 This gives us an e1 of 6 .99.

05:57 Times 10 to the fifth newtons per coulum.

06:01 So there's one of the things we needed to find.

06:05 All right.

06:08 Now for the charge at corner two, we're going to need the length of the diagonal.

06:19 And let's just draw.

06:24 So this is the r that we need, and those are the two legs, and this is a right triangle.

06:32 And so r squared is going to be d squared plus d squared, and so let's look at e2y.

06:46 Okay.

06:47 The other thing we have is that if we take the sign of 45, so let's just remind ourselves what e, those are the 45s, there's e2, e2y.

07:05 So if we take the sign of 45, we get e2y over e2.

07:11 So e2y is e2 times the sign of 45.

07:22 So then e2y is e2.

07:27 E2 is kq over r squared, and then we've got a sign 45 to tack on.

07:35 So e2y is our 8 .99, 10 to the 9, newton meter squared per coulomb squared, we've got the 7 microculems, so 7 times 10 to the minus 6 couloms.

07:53 Okay, r squared is 0 .30 meters squared plus 0 .30 meters, and we're going to square that...

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