Magnetic Flux - Online Tutor, Practice Problems & Exam Prep

Magnetic flux, denoted as φ_b, quantifies the amount of magnetic field passing through a surface. It is calculated using the equation φ_b = B × A × cos(θ), where B is the magnetic field strength, A is the area, and θ is the angle between the magnetic field and the surface's normal. Unlike electric flux, magnetic flux is always positive. The unit of magnetic flux is Webers (Wb), equivalent to Tesla times meter squared (T·m²). Understanding magnetic flux is essential for applications in electromagnetism and related fields.

Magnetic Flux

Hey, guys. How's it going? So in this video, we're going to talk about a concept called magnetic flux. Now, this is going to be extremely similar to how we talked about electric fluxes. We talked about Gauss's law and electric fields. Now this video is going to be extremely similar to that. So, if you need a refresher, go ahead and watch that video again. Let's check it out. So when we talked about electric flux, it was basically just the amount of electric field lines that passed through a surface. So if you have these electric field lines that are passing through the surface right here, then we have 3 variables. We have the amount of electric field lines right here. We also have the area or the size of the surface that it was passing through, so that's the area. And then we also had the angle between them. And the electric flux was basically just the relationship between all of those three variables, e a times cosine of theta. And we also had the units associated. Well, the idea is that magnetic flux is the exact same thing, but instead of the electric field, it's the amount of magnetic field that passes through a surface. So literally this diagram is the exact same. The only thing that's different is that we have a b field or magnetic field instead of an electric field. And so what we do is we just replace these letters or the e with a b. So that means that the magnetic flux, which is given by phi b, is just going to be b times a times the cosine of the angle, where we have the strength of the magnetic field, we have the actual size of the object, and this theta, this angle right here, represents the angle between b and the normal of the surface. So, remember that the normal of the surface is that if I have the back of my hand right here, this pen sort of points in the perpendicular direction. So, in any surface, the normal is always going to be perpendicular to that surface. And if you have that, then this angle basically represents the angle between the b field, which points out this way, and the area vector. So that's going to be right here, that angle right here. Okay? So, the last thing that we need to know is that the magnetic fluxes are always going to be positive, whereas when we talk about electric fluxes, it could be positive or negative depending on whether we're going outside or inside. So magnetic fluxes are always going to be positive, and that's basically it. So let's go ahead and check it out because it's going to be very similar to how we dealt with electric fluxes. Cool?

So what is the magnetic flux through this square surface that's depicted below? So we have the strength of the magnetic field. We have a square surface that's over here. We're sort of looking at it from the side, and we're told what the side length is. So we're going to start off with our equation. The magnetic flux, which is given by phi b, is just going to be b times a times the cosine of the angle, in which this angle represents the sort of angle between b and a. Now if we're looking at this object here on the surface, then the normal vector is always going to be perpendicular to that surface. In other words, it's going to point towards the right. So if you have a surface like this, then that means that the perpendicular vector is going to point outwards like this. You're always also going to assume that it points sort of along the same direction as the b field because then you'll get a positive number there. So you have sort of a choice. You could have written it like this, but we're actually always going to stick to the right or alongside the magnetic field lines because then that would be perpendicular. So there that would be positive. Okay. So we have the electric, we have the magnetic field strength, what it is. And then how about this area? Well, this area right here, the area of a square is just given as side squared. So in other words, if the side length is 5 meters, then the area is just 5 squared, which is 25 square meters. Okay? So we have the area. Now we just have to figure out the cosine of the angle. So we're given this angle of 30 degrees. The problem with this angle is that this is actually the angle between the plane of the surface and the magnetic field. It's not actually the angle between the area and the magnetic field. This is the area, so that means that this right here is the angle theta that we need. So it's not going to be this 30 degrees. What we actually need is we need this angle right here, which is actually going to be 60 degrees because 60 plus 30 would be this right angle right here. Alright? So you have to be very careful with how you choose that angle. So let's go ahead and plug it in. So we've got this phi b is equal to we've got 0.05 Tesla, and then we're going to multiply this by 25, and now we have the cosine of 60. And if you work this all out, you should get 0.625, and the units for that are actually Webers. So I kinda talked I think I forgot to discuss it up there, but that's the unit for these. It's called Webers or that's equal to a Tesla times a meter squared. Alright? So that's it for this one. This is the answer. And, whoops. Let me know if you guys have any questions, and I'll see you oh, man. And I'll see you guys in the next one.

A ring of radius 0.5m lies in the xy-plane. If a magnetic field of magnitude 2 T points at an angle of 22° above the x-axis, what is the magnetic flux through the ring?

• A. 0.59 Wb
• B. 0.69 Wb
• C. 1.46 Wb
• D. 1.57 Wb

Magnetic Flux of a Rotating Ring

Alright, guys. Let's work this one out together. So we're told that you have a radius, some ring of some radius that is in the presence of a magnetic field, and the ring begins with its plane parallel to the magnetic field and then ends perpendicular. We're gonna talk about it just in just a second. We're supposed to be figuring out what is the change in the magnetic flux. Now the keyword in this problem is the change. So in other words, we're looking for some magnetic flux, so that's ϕ_b, but we're looking for the change and we represent that by a delta symbol, remember? So that means that we just need to find out what the ϕ_b final is minus whatever ϕ_b initial is. So we're basically going to have 2 separate sort of initial and final states, and we're going to have to figure out those 2. The initial and the final, I'm going to go ahead and draw diagrams for those 2 right here. So let's say the magnetic field just happens to point to the right. It doesn't really matter where we choose to draw it as long as we're just consistent. Right? We have the magnetic field that points in this direction, and I'm just going to assume that it points to the right like this. We have the exact same magnetic field over here. Now what happens in the between the initial and the final is that the ring is going to rotate like this. And we're told that this ring begins with a plane parallel to the magnetic field. Now what that means is that the actual ring itself, the lines of the ring, the plane of it is parallel to the magnetic field, not the normal. And then what happens is, finally, it ends up with the plane of the ring perpendicular to the magnetic field. So that means it's actually going to be vertical like this. We're supposed to figure out what the change in magnetic flux is. Alright. So we have our ϕ_b final, our ϕ_b initial. So we can go ahead and write out the equations for those. That's Δϕ_b is equal to B·A·cos(θ) but what happens is we know that the angle is changing, whereas B and A are going to remain constant. So these guys don't actually change. These are constants right here. So, let me write that. So, these are constants. And what's actually happening is that we have an angle, so θ final and then minus B·A·cos(θ initial). And these angles right here represent the angle between B and A. So let's go ahead and find out what those are. Let's start out with the final case. Right? So we know that if the ring sort of sits vertically, then that means the plane of it is going to be, or sorry. The normal is going to be perpendicular to that surface. So that means it's going to point out to the right. There's always one we want it pointing alongside the magnetic field. So, what's the angle between B and A? Well, if these things point in the same direction, then that means that this θ here, which is actually θ final, is going to be equal to 0 degrees. Right? Because they point in the same direction. And so, the cosine of 0 degrees is just equal to 1. Now, what that does for our equation is we say, okay. Well, if the cosine of 0 is 1, then that means this term right here just equals 1 and we can sort of just, you know, get it out of there because one doesn't do anything to the equation. So that means this is actually just equal to B·A. Now we have to look at what the initial angle is. So, we have a minus sign right here. Right? So let's look at what the initial angle is. Well, the area is, again, perpendicular to the surface, so it's going to be pointing up in this case because now we have the ring sort of lying flat like that and it's always going to be perpendicular. So this is our area vector and the θ angle represents the area between B and A. So that's actually going to be this guy right here. Now, hopefully, you guys realize that this angle right here, θ initial, is actually equal to 90 degrees. And if you go ahead and work out what the cosine of 90 degrees in your calculator is, you're going to get 0. So what happens is this whole entire term gets wiped out because this is just equal to 0. Okay? So that means that the change in the magnetic field is literally just B·A and then minus 0, because we have nothing there. So, let's just go ahead and multiply those 2 out. So we have a 0.6 Tesla magnetic field, and then the radius is 2 centimeters, so that we have ∏(0.2 ²). And so if you work this out, you're going to get 7.54×10^-4 Weber's. Okay? So that's the change in the magnetic field. Let me know if you guys have any questions, and I'll see you in the next one.

Here’s what students ask on this topic:

What is magnetic flux and how is it calculated?

Magnetic flux, denoted as φ_B, quantifies the amount of magnetic field passing through a surface. It is calculated using the equation:

$\phi$_B=B×A×cos⁡(θ)

where $B$ is the magnetic field strength, $A$ is the area, and $\theta$ is the angle between the magnetic field and the surface's normal. The unit of magnetic flux is Webers (Wb), equivalent to Tesla times meter squared (T·m²).

What is the difference between electric flux and magnetic flux?

Electric flux and magnetic flux are similar concepts but apply to different fields. Electric flux measures the amount of electric field passing through a surface, while magnetic flux measures the amount of magnetic field passing through a surface. The equations are analogous:

Electric flux: $\phi$_E=E×A×cos⁡(θ)

Magnetic flux: $\phi$_B=B×A×cos⁡(θ)

Unlike electric flux, which can be positive or negative, magnetic flux is always positive.

What are the units of magnetic flux?

The unit of magnetic flux is the Weber (Wb). One Weber is equivalent to one Tesla meter squared (T·m²). This unit measures the amount of magnetic field passing through a given area.

How does the angle between the magnetic field and the surface affect magnetic flux?

The angle $\theta$ between the magnetic field and the surface's normal significantly affects the magnetic flux. The flux is calculated using:

$\phi$_B=B×A×cos⁡(θ)

As the angle changes, the cosine of the angle changes, which in turn changes the magnetic flux. When the angle is 0 degrees (field is perpendicular to the surface), the flux is maximized. When the angle is 90 degrees (field is parallel to the surface), the flux is zero.

Why is magnetic flux always positive?

Magnetic flux is always positive because it represents the total amount of magnetic field passing through a surface, regardless of direction. Unlike electric flux, which can be positive or negative depending on the direction of the electric field relative to the surface, magnetic flux does not have a directionality component that would make it negative.