(a) Find the electric field (magnitude and direction) a distance z above the midpoint between tw

step 1 In this prong, we have two point charges, same charge, Q. We're looking for the electric field o

(a) Find the electric field (magnitude and direction) a...

Key Concepts

Electric Field

The electric field is a vector quantity that represents the force per unit charge experienced by a test charge placed in the vicinity of other charges. It is calculated from Coulomb's law, and its magnitude and direction depend on the source charges and the distance from them. Understanding the electric field is essential to analyze the interactions between charges in electrostatics.

Coulomb's Law

Coulomb's law provides the fundamental basis for calculating the electric force and electric field generated by point charges. It states that the magnitude of the electric force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. This law forms the foundation for deriving the electric field from point charges.

Principle of Superposition

The superposition principle in electrostatics states that the total electric field due to multiple charges is the vector sum of the individual electric fields created by each charge independently. This concept is crucial when dealing with systems of multiple charges, as it allows the separate contributions to the electric field to be calculated and then summed to find the net field.

Electric Dipole and Dipole Field

An electric dipole consists of two equal and opposite charges separated by a small distance. The electric field generated by a dipole has a characteristic angular dependence and falls off with distance faster than the field of a single charge. Understanding dipole fields is important when analyzing configurations with both like and unlike charges, particularly in the far-field limit where the arrangement behaves as an effective dipole.

Far-Field Approximation

The far-field approximation is a technique used when the distance from the charge distribution is much larger than the separation between charges. In this limit, the detailed structure of the charge distribution becomes less significant, and the electric field can be approximated by simpler forms, such as those corresponding to a point charge or a dipole. This approximation is valuable for checking the consistency of more general results in regions where z is much greater than d.

Transcript

00:01 In this prong, we have two point charges, same charge, q.

00:06 We're looking for the electric field of point p, which is z, distance c from the origin.

00:12 Each charge is a d over 2 along the x direction from the origin.

00:18 Now, before we get into the electric field, we're going to need sine and cosine of the angle.

00:22 Sin theta is going to be z over r.

00:26 R is basically c squared or d over 2 squared, cosine theta, which is adjacent over hyponous, is going to be d over 2, square root, c squared, plus d over 2, square.

00:44 So that's everything we need now.

00:46 Now, let's talk about the electric field.

00:49 By definition, the force on a unit positive test charge.

00:54 So imagine, you think of it here, but it's not really there.

00:56 That unit positive charge.

00:59 So just look at the force.

01:00 Positive q, positive test charge, repulsion.

01:04 So this would be, this would be, i'm kind of off my angle a little bit.

01:10 This is e1, and this would be, this is e2.

01:18 It's going to be the same magnitude, e1, e2, because of the same charge, same distance.

01:24 So same magnitude, obviously different vectors, but that will take care of in a second.

01:30 So, x component cancel, z components will add.

01:37 So we get e is equal to 2, e1, sine theta, z hat.

01:45 So this becomes 2, 1 over 4, 5, epsilon 0, q over r squared.

01:57 So z squared plus d over 2 squared, and then we got the sign, c over c squared plus d over 2 squared, one half power, c hat.

02:17 So we have first power, half power.

02:21 At them up, three halves power.

02:23 So this becomes one over, let me write it over here, 1 over 4 pi epsilon 0, 2, qz over c squared, plus d over 2 squared, 2, 3 halves, power, z hat.

02:47 So that is the electric field in this case.

02:55 Okay, now we're going to look at when z is much greater than d.

02:59 Or another way of saying it, d over z is much less than one.

03:02 So we're going to use, we're going to use the binomial approximation.

03:07 What is that? one plus x to the n, basically is approximated as 1 plus nx.

03:14 X being much less than 1.

03:16 Obviously, this is a, this is an infinite series, so we're just keeping the first two terms in that.

03:22 That's our approximation.

03:26 So, let me look at just part of this, separated out, the one over the three -half's quantity, and look at that approximation.

03:36 Now, put it in.

03:37 We don't want to ignore anything just yet, but i'm going to write out the approximation.

03:44 So this is the bottom part...

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