FIGURE P23.41 is a cross section of two infinite lines of char...

Question

FIGURE P23.41 is a cross section of two infinite lines of charge that extend out of the page. Both have linear charge density λ. Find an expression for the electric field strength E at height y above the midpoint between the lines.

Cross section of two infinite lines of positive charge with a point at height y above the midpoint between them.

Accepted Answer

Hi everyone. Let's take a look at this practice problem dealing with electric fields. This problem says given the linear charge density lambda for the scenario depicted in the figure below which shows a cross section of two indefinitely long, uniformly charged thin wires along the Z axis. A formula for the magnitude of the electric field E at a height s above the center point situated between these charged wires. The question were given a diagram of what was described in the problem. We're also given four possible choices as our answers. For choice A we have one divided by the quantity of two pi epsilon N. That fraction is multiplied by the quantity of eight lambda s divided by the quantity of D minus four S for choice B, we have one divided by the quantity of two pi epsilon N. And that fraction is multiplied by the quantity of eight lambda divided by the quantity of D plus four S. For choice C, we have one divided by the quantity of four pi epsilon knot. And that fraction is multiplied by the quantity of lambda S divided by the quantity of D squared minus S squared. And Finally, for choice D, we have one divided by the quantity of four pi epsilon knot. And that fraction is multiplied by the quantity of 16 lambda s divided by the quantity of D squared plus four S squared. Now the question wants us to find the magnitude of the electric field due to both lines of charges. So let's draw our electric fields on our diagram. Now in the diagram, the lines of charges are both indicated as being negative charges. So that means the electric field is going to point towards each of those lines of charges. So for the line of choices, it's on the left, I'll have the electric field at my point in between the two charges and a distance s above the midpoint. So that is going to point down into the left towards the line of charge. And likewise for the line of charge on the right, my electro field is going to point down into the right now. The point in question that we're trying to calculate the electric field for is an equal distance from both lines of charges. And since both lines of charges have the same linear charge density, that means the magnitude of the electric field is going to be the same, we label each of those electric fields with um the variable E. So they have the same magnitude. That means that their X components are actually going to cancel out since one is going down to the left and one's going down to the right, that means their X components will cancel out. So when I combine these two electric fields for my total electric field, it's only going to have a Y component and it's going to be pointing downward. Now, the question asked for the magnitude of the electric field. So in order to calculate the magnitude electric field, I just need to take the absolute value of my Y component for the total electric field. So that's what we're going to calculate. So we'll have the absolute value of EY and that's gonna be equal to two E. Since the two Y components will combine together, we have two multiplied by the magnitude E and this is gonna be multiplied by the cosine of theta. And here one labeled theta as the angle that the electric field makes with the vertical axis. And so that will give us just the Y component. Now, we need to recall our formula for the electric field for an infinitely long line of charge. And so recall for that, we have E is equal to lambda divided by two pie. Absolutely not R where lambda is the linear charge density and R is going to be the perpendicular distance from that line of charge to the point in question. Now, we don't want our in our final answer. So we need to write it in terms of the quantities that were given and the problem. And if we look at that, we can use the geometry of the problem. So we'll label R as the distance that is going to be from the point in question to the line of charge to either one. So we'll look at the line of charge on the right and that's gonna be the hypotenuse of a right triangle where one leg is gonna be S and the other leg is gonna be D divided by two. So D here is gonna be the distance between the two lines of charges. And so, since we're looking at the midpoint, that leg is just gonna be D divided by two, that means that R is going to be equal to the square root of the quantity of D divided by two in quantity squared plus S squared. Let's just use some Pythagorean's theorem. So we can plug that into our formula for the electric field and we'll have E equal to lambda divided by the quantity of two pi epsilon knot multiplied by the square root of the quantity of D divided by two quantity squared plus S squared. Now, we also want to write cosine data in terms of the given quantities and the problem. And we'll again use geometry. So if I look at that same right triangle, I'll have cosine of data, it's going to be equal to S divided by R. But we already found an expression for R and it's gonna be S divided by the square root of D divided by two quantity squared plus S squared. So now I have both of these expressions, I can plug them back in for um our equation for the magnitude of the electric field. So we'll have the absolute value of ey gonna be equal to two, multiplying the quantity of he will substitute in our expression for E. So they'll be lambda divided by two pie. Absolutely not multiplied by the square root of the quantity of D divided by two in quantity squared plus S squared. And that's gonna be multiplied by our expression for cosine data, which is S divided by square root of the quantity of D divided by two quantity squared plus S squared. So at this point, we just need to simplify some things. So we'll have the absolute absolute value of EY is going to be equal to here. I'll just combine the um expressions into a single expression. So we'll have two lambda s divided by two pie. Absolutely not multiplying the quantity. When I multiply the two square roots together, the square root goes away. And so I'll have the quantity of D divided by two quantity squared plus S squared. Now, what I'd like to do now is get rid of the fraction that's in the denominator there. So I'm going to multiply both the numerator and denominator by four. And so we'll have the absolute value of EY is equal to eight lambda s fight by two pie. Absolutely not multiplying the quantity of D squared plus or S squared. And if I look at my answer choices, this doesn't really match up to any of my answer choices. So to get it to match up with the answer choice, I'm gonna multiply our expression in uh the form of one, in the form of two divided by two. So when I do that, I'll get ey is equal to. And here I'm gonna pull out a one divided by four pi A not. And that's gonna be multiply in the quantity of 16 lambda s divided by D squared plus or S squared. So now when I look at my answer choices, this actually corresponds to answer choice D. So just to recap of what we've done here, we start off by finding the total electric field and noticing that from the sentry of the problem, the X components canceled out. And so we just had to look at our Y component for the electric field. And so we calcul calculated that using our formula for the electric field due to an infinitely long line of charge as well as the geometry of the problem. And once we did that, we just had to simplify expression and get it to match up with one of our answer choices. So I hope that this has been useful and I'll see you in the next video.

FIGURE P23.41 is a cross section of two infinite lines of charge that extend out of the page. Both have linear charge density λ. Find an expression for the electric field strength E at height y above the midpoint between the lines.

Key Concepts

Electric Field Due to a Line of Charge

Superposition Principle

Coordinate System and Geometry

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