In this video, this is your question number two for the EOT. So in this video, we are going to learn about electric flux. Define the electric flux.
Okay, this should be the first part. How we need to define? as a dot product that is the vectors between the home electric field and the area means there should be a dot product between dot product between electric field and area.
Clear? Okay. At each point of the surface and express it that in an equation.
Okay. This is your equation for the flux. Okay. Second part. Prove that electric flux.
We need to prove mathematically. Prove that electric flux passing through a closed surface. We have the two type of surface.
One is the closed surface and one is the open surface. Is given by by net charge okay q inside this is the keyword huh this is the keyword because whenever we are solving any question of ghost theorem within the system and outside the system both are different huh inside the surface divide by okay divide by permittivity so what does it mean that charge divide by permittivity that should be equals to flux okay we need to prove that huh so i will prove this as a mathematical relation of the medium okay we need to use as a medium also epsilon and epsilon naught write the goes law in its integral form integral form means closed system apply goes law to relate the net flux through the closed surface i told you now if we are talking about integral form it means closed surface And if we are talking about normal form, that is the open surface. Okay.
So, guys, in this part, we have the two things. First thing, the electric flux. What exactly the electric flux is? What is the definition?
And how can you find out the maximum and the minimum values? After that, we are going to learn the proofs. And then third part, we are going to solve the numerical. So, guys, I'm not starting the numerical first.
I'm going to start the basics first. So that you're going to understand. understand i already write down the definition for you and after definition we will cover the things so what is the definition of electric flux so electric flux says what is the electric flux electric flux says the electric field lines electric field lines passing through a given area let's say you have any particular area and how many line of forces are passing through it is called flux let's see example. Let's say this is a surface.
Any defined surface and how many line of forces how many line of forces these are the Electrical line of forces how many electric line of forces are passing from this area? This is called flux. So this is your electric field and this is your area This is your electric flux. There are two cases case one and case two Okay, the case one is your open surface Let's say this is your open surface open Surface and we have second case for closed surface.
Let's say you close something and we are talking about this area okay electric field lines of forces are going out so closed surface and open surface so in case of open surface we are going to directly use this equation so where this phi is called flux this is the symbol of electric flux this is your electric field e is your electric field and a is the area and theta is the angle between them okay and second formula that is for the closed surface here we are using this part i will just read i will write again for you what is this this is the integration first time integration second time and there is a circle circle means this means closed this means closed okay now what we are going to solve the maximum and the minimum value and we are going to learn where is the angle theta okay just check before going to learn any part from the ghost here we need to learn about the quantities the tensor quantity tensor Tensor quantity. What is the meaning of tensor quantity? Tensor quantity means when a scalar, when a scalar, we give the direction, scalar plus given direction, given or associated direction. that is called tension tensor so in this the direction is always always outward just remember this thing always outward normal outward normal I will show you how to solve this part let's say this is a area okay this is called area So do you know any direction for area?
Generally it's a scalar quantity. Yes or no? Scalar quantity. But now we are providing him this area a direction. So area with direction.
and I told you outward normal outward normal okay so how to show that let's say this is the area so just see let's say let's say you are you can see a paper so what should be the direction that direction should be outward towards yourself and it is represented with the help of n cap so if i write a vector area vector area vector so it should be written as area multiply n cap you remember now how to write a simple vector let's say vector b how will you write it b b cap okay in this Direction is along B. But here in case of area because it's a tensor quantity. So area I can write like A vector which is equals to AN cap.
It is given direction. Given direction. Clear?
This is the difference. This direction B cap is associated with B vector. Okay? Not But here in tensor, we are giving the direction, n cap, that is outward.
So I will show you two cases. I will explain two cases to you so that you can understand in a better way so that we can solve the Gauss theorem. So the first case, part 1, let's say you have the surface, okay? Then outward normal is like this. Clear?
Okay, case 2. If the paper is like this, you can check with your mobile. let's say or hand okay so in that case what is the outward this is your outward normal okay so i can write a vector is equals to a this is your magnitude and this is your n cap fine okay now guys we are going to solve two parts maximum and the minimum maximum okay i'll make it big maximum and minimum flux okay we know the formula Phi is equals to E A cosine theta. Flux is equals to E A cosine of theta. First see the diagram.
Let's say we have this diagram. We have this surface. Let's say this is the surface. And the case 1. You can solve easily.
Case 1. Because we know one thing is straight. The N cap is going outward. N cap is going outward.
Let's say electric fields are going like this. Electric field is going like this. These are the electric field lines.
Okay. So, can I say electric field vector is equals to E multiply E cap. Fine.
So, can we easily. find out what is the angle between them the angle is 90 degree how electric field lines e cap is going like this n cap is going like this so what is the angle 90 degree clear case 2 so that you can solve easily now case 2 let's say the same I'm just going to change the shape of the surface let's say surface is like this Okay, and electrical line of forces are moving like this. This is the line of forces Electric field so can I say e is equals to e? times e cap fine okay now tell me what is the direction of n cap so can i say n cap is here outward normal so can we easily say the angle between e cap and n cap so the theta is equals to zero clear so if you solve this first part now and the second part so we know phi is equals to e vector dot a vector okay now solve them properly huh so that you can understand can i write e vector e vector is equals to e times e cap and a vector as a times n cap so write it so e times e cap dot a times n cap so take out the magnitude value e And A. Both are magnitude bracket. E cap dot n cap.
So what is the angle between them? So can I say E A. Angle between them is 0 degree. Theta equals 0 degree.
So E cap n cap cos of 0. No. In this case angle is 90. First case angle is 90. So cos of 90 degree. And cos 90 is 0. zero so the flux is zero whereas flux value is zero cos 90 degree is equals to zero fine okay now case number two we know flux is equals to E dot a. You can just solve directly this one.
Now I'm going to use it directly from equation 1. Can I say e a bracket e cap n cap. of 0 degree so in that case we get the maximum value ea and you need to remember e cap is direction so what is the value 1 n cap is your direction so what is the magnitude 1 so what we get ea which is your maximum value clear guys you understand the zero value this is your zero value okay and this is your maximum value and so now easy to understand whenever your theta is 90 degree or I'll start with 0 degree. So whenever your angle is 0 degree, flux is maximum. Okay, whenever your theta is 90 degree, flux is zero and whenever your theta is 180 degree flux is minimum or you can say the negative maximum or which is equals to minimum value clear guys okay now till now we have learned about about the maximum minimum values and about the open values and the all these things clear now what we need to understand the proof how you can prove that goes theorem is equals to okay now just go back to the question so that you can understand now now just read this what we need to prove that that the electric flux that we need to prove electric flux which we know integration over a closed path e dot d a we need to prove that q divided by epsilon naught read the question prove that electric flux you Electric flux. Through a closed surface.
I told you this is the equation for closed surface. Clear? Open surface is different. Closed surface is different.
Given by net charge. So Q net. Clear? Inside the surface. Inside means close the surface.
Divide by permittivity. This is your permittivity. Clear?
Of the medium. And right goes low in its integral form. So this is your integral. integral means with the help of integration. Clear?
We need to prove that. Okay. Now, proof of Gauss theorem.
Gauss theorem. So, guys, what we need to prove? Flux equals integration over a closed path.
This means integration or integral. Integration over a closed path E dot DA should be equals to Q net divided by epsilon naught. Clear? Okay.
Now it's very easy guys. Just take any surface. Okay. Consider.
Consider a surface. Let's say this is a sphere. What a shape of my sphere, huh?
Okay, let's say now this will be fine. Okay, let's say this is a sphere, huh? so in the sphere solve it so let's say this is the closed surface and from here we are finding the value this is your n cap this is your e cap and solve the value This is your E-CAM. So we know LHS. So flux is equals to integration over a closed path E dot DA.
Clear? Okay. Now open this. Integration over a closed path.
E dA cos of theta. We already understand this part. Okay. And we understand both are in the same direction. E cap.
N cap. So theta equals to 0 degree. So can I say when theta equals 0 degree, cos of 0 degree is equals to 1. So cos of 0 is 1. Okay. Let me add a paper.
Okay, now, so we have integration over a closed path E, D, A. So, we know the value of E. What is the value of E, guys? K times or you can say 1 over 5. 4 pi epsilon naught, 1 over 4 pi epsilon naught, q over r square.
Fine. Place the value. And what is the integration of A? The flux is equals to E double integration. of DA.
Okay. So what is this part? Solve it.
Integration of DA is the area. So we are using a sphere. So what is the area?
4 pi r square. So use the value flux is equals to E is 1 over 4 pi epsilon naught, 4 pi epsilon naught Q over r square. Multiply.
What is the area? Area 4 pi r. square so cancel this 4 by 4 by R square R square so what is left flux is equals to Q over epsilon naught and which is the required clear guys required that we need to prove that what exactly we are proving flux is equals to integration over a closed path e dot d a which should be equals to q by epsilon fine guys okay now we are going to solve the numerical.
Numerical is just an easy part and you can solve in an easy way. Okay. Now you will understand the numerical also.
Okay guys, just read the question what exactly it is saying. We have a cube here. In the cube, how many faces we have?
We have the six faces. So we need to solve the flux through all these six faces. Now just try to understand.
Let's say I'll draw a new cube for you for the better understanding. Okay, we have this cube. Fine.
And solve this cube. How many faces you can see? Shall I write the numbers? Okay, I will write the numbers.
Let's say one face, two face, up face three, down face four, front face five, six, and five. Okay, you can understand now all the six faces. Okay, now what he is exactly telling us.
Show that cube has the faces of area A. All are same area. Uniform electric field means E remains same. E is perpendicular.
E is. perpendicular to the plane of one face of the cube what is the net electric flux passing through the cube guys what is the net flux this is the major point net electric flux how many faces simple question very simple how many faces you Six faces. So means we need to find six flux. Total flux should be equals to phi 1 plus phi 2. plus phi 3 up to 5, 6. Okay? So, you need to find a phi.
So, what is the direction of electric field? Direction of electric field. So, what is the direction? Like this.
Can I say in terms of positive x-axis? Yes or no? From here, you can easily understand.
Let's say I'll draw the... You can see from here, this is along the positive x-axis. Okay, now just check.
Okay, let's see. This is the box and you can understand from here. It is passing like this. So, which area it is passing?
as per the surface take the surfaces huh third surface take the third surface where is the area can i say here is the area n cap yes or no where should be the fourth area wise huh let's say area 1, 2, 3, 4, 5, 6. Okay, just tell me. For area 1, what is the end cap? End cap is outward. N cap is outward. Can I put here N cap?
This is for area 1 and this is the electric field. Clear? So can I say theta is equals to 0 degree?
Cos 0 degree is 1. so flux 1 is equals to e a cos 0 which is equals to flux is equals to e a clear guys easy okay now for second area So, where is the n cap in second area? Okay, it should be this side, n cap. So, guys, n cap outward, okay. But what is the angle? Can I say theta is equals to 180 degree?
See, try to understand. It is going this side, n cap this side, electric field this side, clear? So, theta is 180 degree.
So, when you solve cos 180 degree. then what will you get flux is equals to e a cos 180 so you will get flux is equals to negative e a fine now third four five six just check the third one again check it properly out where the line of forces are going end cap this side and electric field this side so what is the angle see n cap is going up. Electric field is going like this. Can you see what is the angle we have? We have the angle of 90 degree.
Clear? In case of 4 also. N is going down. N is going down. E is going like this.
Can you see the angle 90 degree? Okay. Case 5 also.
You can see front and back. So it is making an angle of N and E cap. making an angle of 90 degree. So, all the four cases, we have flux is equals to Ea cos of 90. Cos 90 degree.
is 0. Can I say flux is 0? Clear guys? So we have all the 6 values, flux 1, flux 2, flux 3, flux 4, flux 5, flux 6. So just add them up and finalize it.
So flux is equals to flux 1 plus flux 2 up to flux 6. Put the values. What is flux 1? E, A. What is flux 2? negative of EA flux 3 0 flux 4 0 flux 5 0 flux 6 0 so what is the net value of this 2 plus EA and minus EA so flux is 0 it means the line of forces see this is the major last point is the difficult most or the most you need to focus on what is the conclusion Conclusion is line of forces or electric field in equal electric field out.
What does it mean? Let's say this is the body. say 10 line of forces 10 electric field is going inside the surface and how many is going out 10 electric field line of forces are going out means inside you have nothing clear so this is the real understanding so there is no net flux okay guys