The cube in FIGURE EX24.7 contains negative charge. The electr...

Question

The cube in FIGURE EX24.7 contains negative charge. The electric field is constant over each face of the cube. Does the missing electric field vector on the front face point in or out? What strength must this field exceed?

Cube with electric field vectors showing strengths of 10 N/C on three faces and 20 N/C on two faces.

Accepted Answer

Hello everyone. Let's go through this practice problem. Given a cuboid shaped room with uniform electric fields across all faces. A negatively charged object is placed at the center of the room. As shown the back face of the room has a missing electric field vector, determine the direction of the missing vector inward or outward and calculate the minimum strength needed to maintain uniformity inside the room. Option A inward at 10 newtons per Coolum, option B inward at 20 newtons per Coolum, option C outward at 20 new ones per Coolum and option D outward at 10 new ones per Coolum. OK. So right off the bat, there are a few important things we need to realize. First off the problem tells us that the room has a negatively charged object inside. Recall that according to convention, electric field lines point towards negative charges. So if we have a net, if we have a net negative charge inside the room, then there should be a net inward flux for the room because there is a net flow of field into the room towards those negative charges. So this means that the total flux, it would be less than zero. And we need to find the condition for the back flux that satisfies this condition. And looking at this diagram, we can see that there are some parts of flux that are pointing out of the cube, some that are pointing into the room. So there are a few different things there to keep an eye on. So to expand this condition out a bit, we'll say that the flux inwards. So the in all the inwards fluxes into the room plus all the outward flexes that were given plus the one for the back specifically which you want to find should all add up to be less than zero. So again, this is the condition we're looking to satisfy and we want to solve for the back flux. So the first thing we should do really what we should be trying to do is finding formulas for the inwards and outwards fluxes that were given and then use this equation, use this condition to solve for the back flux. The next thing we want to remember is the formula for electric flux because the only new, the only numbers the problem gives us are the values for the electric field strength itself. Recall that an electric flux is given by the electric field strength multiplied by the surface area multiplied by the cosine of the angle between the direction of the field through that surface and the vector normal to the face of that surface in a lot of cases though like this one, we can assume that that angle is 90 degrees, which will be the case if the, if the electric field is going directly into like straight into the surface at a 90 degree angle, so the angle becomes 90 which means the cosine term just becomes a one. So in most cases, we can use this simple formula where it's just the electric field multiplied by the area. So let's write out a formula for the net inward flux from the vectors we're given. Looking at the diagram, we can see that there are three faces that have an inward flux, that's 24 and five. So we will add them up and since they're in words, they will be negative, so that's negative E sub two multiplied by A minus E sub four multiplied by A minus E sub five multiplied by A since it's a cube and all the A's are the same, we can simplify this by factoring the a out outside of parentheses. So we can write this as a multiplied by and then in parentheses and then we add up the electric fields. So E sub two according to the diagram is 20 newtons per Coolum. So 20 newtons per Coolum uh minus E sub four which is 10 new ones per Coolum minus E sub five, which is 15 new ones per coon or doing the math on this. This is just equal to negative 45 new ones for coon all multiplied by A. So now we have an expression for the inward flux that we're given. Now, we'll do the same thing before the outward flux or the outward flux. And we can see from the diagram that there are two fluxes given that are outwards which is one and three, they're both pointing away from the center of the cube. So we'll use the same formula except now the flux, the flux components are both going to be positive because they're outwards and we'll use the same idea. It's gonna be the surface area of the cube multiplied by E sub one plus E sub three. And of course, those values are 15 newtons for Coolum and 10 Nuances forum or more simply that is 25 five newtons per Coolum all multiplied by the surface area. So now that we have flux in and plug flex out, now we can put these into the condition that we established earlier in the video. So a flux in is negative 45 and newtons per Coolum times A then plus flux out which is 25 newtons per Coolum times A. Then plus the one we're trying to find the flux of the back, which is e back, the electric field of the back again multiplied by A and this is all less than zero. We can simplify this a bit because every single term has the A. So they can all go away and be canceled out. So we just have the electric field terms and then simplifying this down further and doing the math on negative 45 plus 25 you can see that it's negative 20 co uh newtons for coon, I have 20 newtons Pakula plus E sub back in order for this condition to be satisfied. Now, in order to find the minimum condition for this to be satisfied, we instead treat the less than sign like an equal then sign. And we can see that in order for this, for the left side of this equation to be equal to zero E sub back has to perfectly balance out the negative 20 newtons per Coolum will a positive 20 newtons per Coolum. So what this means is that the minimum strength needed to maintain uniformity in the room is 20 newtons per Coolum. And since it's positive, it's also going to be outward. So that means that this is the answer to the problem. The answer to the problem is positive 20 newest for Coolum our word. And if we look at our multiple choice options, we can see that this agrees with option C. So option C is the answer to this problem and that is it for this video. I hope it helped you out and please consider checking out some of our other tutoring videos which will give you more experience with these types of problems. That's all for now. And I hope you all have a lovely day. Bye bye.

The cube in FIGURE EX24.7 contains negative charge. The electric field is constant over each face of the cube. Does the missing electric field vector on the front face point in or out? What strength must this field exceed?

Key Concepts

Electric Field

Gauss's Law

Direction of Electric Field Vectors

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