### Video Transcript

Point charges π one equals 0.0050 nanocoulombs and π two equals 0.020 nanocoulombs are fixed at π one equals 9.00π plus 3.50π meters and π two equals 6.50π minus 2.00π meters, respectively. What is the magnitude of the electric force on π two due to π one?

So we have these two charges: π one and π two. And letβs start out by drawing their position relative to one another. Working with this π₯, π¦-coordinate set, π one is located at nine π and 3.5π, which puts that charge right there as weβve drawn it. π two on the other hand at 6.50π and negative 2.00 π is right there. And we can draw in the distance vectors π one and π two for each of these charges.

Now, in this question, we want to solve for the magnitude of the electric force on π two due to π one. And since weβre speaking of magnitude, we could equivalently solve for the force on π one due to π two; theyβre the same. We can now recall Coulombβs law, which tells us the electrical force that acts between two point charges π one and π two. Itβs equal to the product of those charges times Coulombβs constant π all divided by the distance between the two charges squared.

Now, hereβs a question: whatβs the distance between π one and π two, our two charges? If we call that distance π, we know itβs equal to the magnitude of π one minus π two. If π were a vector, it would look the way weβve drawn it on our graph.

Hereβs how the distance π fits into the overall expression for the magnitude of the electric force on π two. The magnitude of that force, which weβll call simply πΉ, is equal to Coulombβs constant times the magnitude of π one times π two all over π squared. Coulombβs constant π is approximately equal to 8.99 times 10 to the ninth newton meter squared per coulomb squared.

Knowing that as well as π one, π two and π one and π two, weβre ready to plug in and solve for the magnitude of πΉ. What weβve done here is weβve plugged in our values for the numerator and left are denominator for now in terms of π one and π two.

Before addressing that denominator, letβs consider the way weβve written our numerator. We see weβve plugged in our value for Coulombβs constant π. Then, we converted our charge π one from the way it was expressed in the problem statement as 0.0050 nanocoulombs to a number in scientific notation, 5.00 times 10 to the negative 12th coulombs. Weβve done the same thing for our charge π two, moving it from 0.020 nanocoulombs to 2.00 times 10 to the negative 11th coulombs. So thatβs or upstairs.

But what about our downstairs, the denominator? We said that π, the distance between π one and π two, is equal to the magnitude of the vector π one minus π two. And since Coulombβs law calls for this distance to be squared, weβve squared this magnitude term. But how do we actually go about calculating the magnitude of π one minus π two when π one and π two are vectors?

We can say that if π is equal to the magnitude of the difference between two vectors π one and π two, then thatβs equal to the square root of the difference between the π₯-component of these vectors squared plus the difference between the π¦-component of the vectors squared. This is all assuming of course that π one and π two are two-dimensional vectors, as they are in our case.

We can use this expression in order to solve for magnitude of π one minus π two in our case. When we plug in for the given values of π one and π two by their components, we see that the magnitude of π one minus π two is equal to the square root of 9.00 minus 6.50 squared meter squared plus 3.50 plus 2.00 squared meter squared. Then, we notice that this entire expression in the square root sign is itself squared. So that means that the square and our square root sign cancel one another out. This means our denominator simplifies to 2.50 squared plus 5.50 squared meter squared.

Before we enter this expression on our calculator, take a look at the units. Notice that the units of meter squared in the numerator and in the denominator cancel out, as do the units of coulomb squared. So weβre left with units of newtons, which indicates weβre solving for a force. We are! So thatβs a good sign that weβre on the right track.

Entering this whole expression on our calculator, to two significant figures, we find a result of 2.5 times 10 to the negative 14th newtons. That is the magnitude of the electric force on π two due to π one.