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Electric Charge | 15 mins | 0 completed | Learn |

Charging Objects | 7 mins | 0 completed | Learn |

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Coulomb's Law (Electric Force) | 56 mins | 0 completed | Learn Summary |

Electric Field | 46 mins | 0 completed | Learn Summary |

Parallel Plate Capacitors | 16 mins | 0 completed | Learn |

Electric Field Lines | 13 mins | 0 completed | Learn |

Dipole Moment | 7 mins | 0 completed | Learn |

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Electric Flux | 19 mins | 0 completed | Learn Summary |

Gauss' Law | 24 mins | 0 completed | Learn Summary |

Concept #1: Electric Flux

**Transcript**

Hey guys. In this video we're going to be talking about electric flux. Electric flux is going to be really important in the next topic, we talked about which is called Gauss's law. So, let's make sure that we follow this closely and get it, right? Flux in general is a measure of how much of a particular field passes through a surface, okay? Electric flux which is what we're going to be talking about it's a measure of how much the electric field passes through a particular surface. Now, what do I mean by, how much of a field passes through a surface? Well, let's consider these three figures given here, okay? Now, in the first figure, well, in all three figures, we have the same four field lines.

In the first figure, we have a surface, that's vertical like this so the field lines are passing directly into that surface and you can see that all four field lines pass through the surface, okay? So, this surface is effectively caught all four field lines. Now, what about the second surface? Well, the surface is flat and all four of the field lines actually pass over the surface, this one catches, right? catches, none of them, they all pass over or under that surface, what about the third video where the surface is at an angle, it's not up, it's not flat but it's somewhere between? Well, in this case it doesn't catch the upper or the lower field lines but it catches the middle two. So, this will catch all or the maximum number, more precisely, this will catch none and this will catch some, okay? So, the electric flux clearly depends upon it the angle between the surface and the electric fields. Now, what angle are we talking about? This is really, really important to keep in mind and this is a place where a lot of students mess up, it's the particular choice of angle. Now, there's something that we can define which is called the normal to any surface, what the normal is, is it's the perpendicular direction, normal equals perpendicular, I'm having a terrible spelling day, perpendicular, there we go, got it out, okay? The normal means the perpendicular, this was what the normal force meant, normal force is always perpendicular to a surface the normal Direction is also always perpendicular to the surface. So, imagine that my hand was some surface this pen would represent the normal direction and no matter what angle my hand is, that pin will point normal to my hand, okay? So, the angle that we care about here, theta is actually the angle between the normal and the electric field and that's very, very important, okay? Now, what is the actual flux mathematically? Well, it's E, the magnitude of the electric field times A the area of the surface times cosine of theta, or once again, theta is the angle between the electric field and the normal of the area and very, very important to get that, right? The last point I want to make in this video is that this whole flux through any closed surface is the sum of the fluxes that make up that closed surface. So, I mentioned closed, a closed surface, what is a closed surface? A closed surface is any sort of object, okay? A really easy one is a rectangular prism, okay? Or a cube which is a type of rectangular prism but a closed surface is the three-dimensional object but this closed surface is made up of a bunch of two-dimensional surfaces, okay? This equals six rectangles, right? Each of those surfaces are going to capture some amount of flux, some amount of the field lines and the total, the total flux through that entire object or that entire closed surface is just going to be the sum of the flux captured by each of the individual services, okay? Straightforward, let's do a quick and very simple example illustrating this point.

The electric flux through each surface of a cube is given below, what is the total flux through the cube? It's simply the sum, the total flux is going to be the flux1 plus flux2 plus3 etcetera up to flux6 going to be the addition of all of them, this is 100 plus 20 plus 0 plus 0 minus 40 minus 80, okay? 100 plus 20 is 120, 40 plus 80 is also 120. So, 120 minus 120? 0 the total flux through this cube is 0, okay? And that wraps up our discussion on electric flux, thanks for watching.

Example #1: Flux Through Angled Surface and Cube

**Transcript**

Hey guys. We want to do a couple examples using electric flux now. What is the electric flux through the surface depicted below, note that the area the surface is 1 square meter and the electric field is 100 Newtons per Coulomb, okay? So, the electric flux, as we know, is just E the magnitude of the electric field times a there and surface times cosine of theta, where theta is the angle between the electric field and the normal to the surface, okay? This angle is not Theta, this angle is the angle between the electric field and this surface itself, we don't want the angle between the electric field and the surface, we want to angle between the electric fields and the normal to the surface, the perpendicular to the surface, okay? So, if I draw this, we have this angle 30 degrees, the normal is this angle right here, and those just need to add up to 90 so this is 60 degrees because 60 plus 30 is 90, okay? So, the flux is as simple as 100, the magnitude of the electric field, times 1, the area, times cosine of 60, cosine of 60 is 1/2. So, what's 1/2 of 100? 50, okay? And the units are Newton meters squared per Coulomb but we don't really need to worry about those, the number is 50, okay? Moving on to the next example. A cube of side length 2 centimeters is placed in an electric field of magnitude 100 Newtons per Coulomb as shown below, what is the electric flux through each side of the cube, okay? Before we even begin I just want to note that only two of these surfaces will capture any flux, the front surface ,the top surface, the bottom surface and the back surface all have electric field lines that run over the surfaces or run along the surfaces, those surfaces will not capture any flux because they aren't angled into the field lines at all. Remember, those field line needs to pass through the surface not along the surface, so the only two surfaces they're going to capture any flux are the left surface and the right surface. So, let's figure out what those fluxes are, okay? Now, before we should do, before we do that, we should figure out where the angle theta is between the electric field and the normal, in this case the normal is pointing out here and that angle is clearly 0, they're both running parallel, here the normal is also pointing out and in this case, because the electric field is moving to the right, they're pointing in opposite directions so the angle is 180. Now, we can figure out exactly what the flux between those two surfaces is, the flux on the right surface is E which is 100 Newtons per Coulomb times A, okay? A is going to be 2 centimeters times 2 centimeters which is 4 square centimeters which is going to be 4 divided by 10,000 square meters times cosine of 0 degrees and cosine 0 degrees is just 1, so this is 4 times 10 to the negative 2 and the units are Newton meter squared per Coulomb, that's the flux through the right surface, the flux to the left surface is also going to be 100 times the area times cosine, but now 180 degrees, and cosine of 180 is negative 1, not 1, but negative 1, so this is negative 4 times 10 to the negative 2 meter squared per Coulomb, okay? Now, what we want. So, sorry, what we want is the flux through each side of the cube. So, we also have that the flux through the top equals the flux through the bottom equals the flux through the front equals the flux through the back which are 0. So, that, that and that, those are the fluxes through all of the surfaces. Now, the thing here to note, because this will be important later, is the total flux is actually 0, okay? Because the flux coming in, sorry, this is the flux coming in, the flux coming in equals the flux coming out so the total flux is 0. Now, by convention, flux that's entering an object is typically negative and flux that's exiting an object is positive, that's just a sign convention but the important thing here to take away is that whatever flux is entering this object is the same amount of flux that's exiting the object so the total flux is 0, okay?

Example #2: Flux Through Spherical Shell due to Point Charge

**Transcript**

Hey guys. Let's do another example of electric flux. What is the electric flux through a spherical shell of radius r due to a point charge k at the center, okay? All we need to do is figure out, what E, A cosine of theta is. Now, where does E points? Well, E always points outward due to some positive charge q, right? Well, where does the normal of the surface point? The normal of the sphere also always points outwards, it's going to point out right there, it's going to point outward there, is going to point outward there, is going to the point outward there, that's because the outward direction it is the perpendicular direction of a sphere. So, this means everywhere on the surface of the sphere cosine of theta is just going to be cosine of 0 which is 1, this is because the electric field and the normal are always parallel, okay? So, what's the electric field due to a point charge at some radius R? Well, that's just K, q over R squared what's the area of this sphere of radius R okay, if you don't remember, the surface area of a sphere is 4pi, R squared so this is going to be 4pi, capital r squared, so the flux is just going to be k, q over R squared times 4pi, R squared, those R squares are going to cancel, that's an 84pi k, q, okay?

Practice: The electric flux through each surface of a cube is given below. Which surfaces of the cube does the electric field run parallel to?

Φ_{1} = 100 *Nm*^{2} /*C* Φ_{4} = 0 *Nm*^{2} /*C*

Φ_{2} = 20 *Nm*^{2} /*C* Φ_{5} = −40 *Nm*^{2} /*𝐶*

Φ_{3} = 0 *Nm*^{2} /*C* Φ_{6} = −80 *Nm*^{2} /*𝐶*

Practice: What is the total flux through the two surfaces depicted in the following figure? Note that surface 1 has an area of 50 cm^{2} and surface 2 has an area of 100 cm^{2} , and E = 500 N/C.

A uniform electric field with a magnitude of 8x10 6 N/C is applied to a cube of edge length 0.1m as shown in the figure. If the direction of the E-field is along the +x-axis, what is the electric flux passing through the shaded face of the cube?
A) 0.8x104 Nm2/C
B) 8x104 Nm2/C
C) 80x104 Nm2/C
D) 800x104 Nm2/C

A flat sheet of paper of area 0.50 m 2 is oriented so that the normal to the sheet is at an angle of 60° to a uniform electric field of magnitude 14 N/C. Find the magnitude of the electric flux through the sheet.

A charge of 30 C is located at the center of a cube. The electric flux through the right side of the cube is _____________.
a. 5.65 x 1011 Nm2/C
b. 3.38 x 1012 Nm2/C
c. 0 Nm2/C
d. Unable to be determined without more information

Find the flux ΦE of the electric field E = (24î + 30ĵ + 16k̂) N/C through a 2.0 m2 portion of the xy plane (k̂ is the normal).
[A] 32 N • m2
[B] 34 N • m2
[C] 42 N • m2
[D] 48 N • m2
[E] 60 N • m2

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