**Concept:** Magnetic Flux

Hey guys, in this video we're going to talk about the concept of magnetic flux. We've seen in the past, when talking about electrostatics, the electric flux through a surface. Now, we want to talk about a very, very similar concept, the magnetic flux, let's get to it. Now, remember guys, the electric flux is just the amount of electric field lines passing through a surface, identically magnetic flux is just the amount of magnetic field lines passing through a surface, okay? So, for instance, if we have one, two, three, four field lines, either electric or magnetic fields, because the concept is identical and I draw two loops, I draw a loop like this, the red loop, and I draw a loop like this, a black loop, the black loop captures four of these few lines, the red loop captures two, so the red loop has less flux than the black loop, okay? Now, specifically, magnetic flux is given by an equation identical to that electric flux, but the electric field is replaced by the magnetic field. So, we have B, A cosine of theta, the most important thing to remember is that theta is the angle between B and normal to the surface, okay? If you look at this figure right above me, theta, this angle right here, is angle between the magnetic field lines and the normal to the surface, okay? Now, magnetic flux changes are actually going to be really, really important in the context of electromagnetic induction, we didn't really talk about electric field, electric flux changes, but we discussed electric flux in Gauss's law, but now we definitely need to address the change in the magnetic flux, if you look there are three variables in this equation, there's B, there's A and there's theta, so the flux changes with magnetic field, the magnetic field strength you could say, the area of the surface and the angle between the normal to the surface and the magnetic field. One quick thing that I forgot to address, probably because my head was in the way, is the units of magnetic flux are Webers, and that is 1 Tesla meters squared. Alright, it's important to remember this fact, that the magnetic flux changes with a change in magnetic field area or angle, like I said in the context of electromagnetic induction, magnetic flux changes extremely important, okay? Let's move on, let's do a quick example. What is the magnetic flux through the square surface depicted in the following figure if B equals 0.05 Tesla? Assume the side length of the square is 5 centimeters, so they say that this surface is a squared surface, we know the flux, which is what we were asked, which is we were asked for, magnetic flux, is going to be B, A, cosine theta, they told us what B is, right? So, we know that, we have some information about the angle, so we know that, what we need is the area, they told us this is a square, so we know the area is just length squared, whatever the length of that square is, and it told us the side length is 5 centimeters, so it's 0.05 meters squared, which is 5 times 10 to the negative 4 square meters, okay? That's the area. Now, we know everything, where I put a check here, just solve for the magnetic flux, this is going to be a 0.05 Tesla times 5 times 10 to the negative 4 meter squared the area, oh! I did make a mistake here, actually, this is 2.5 times 10 to the negative 4, sorry, 25 times to the negative 4, which is 2.5 times 10 to the negative 3, square meters, 2.5, this is 10 to the negative 3 and cosine of the angle, the question now is, what's the angle? We're given 30 degrees, but we have to keep in mind that theta is the angle between the normal and the magnetic field, this is the normal direction right here.

So, this is the angle that we care about, if this angle right here is 30 degrees, then 30 degrees plus theta has equal 90, right? This right here is the right angle, 30 plus theta has equal 90, so theta is actually 60 not 30, and this whole thing becomes 6.25 times 10 to the negative 5 Webers, okay? And, that's the answer. This wraps up our discussion on magnetic flux and how to calculate it, thanks for watching guys.

**Problem:** 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?

**Example:** Magnetic Flux of a Rotating Ring

Hey guys, let's do an example. A ring of radius 2 centimeters is in the presence of 0.6 Tesla magnetic field. If the ring begins with its plane parallel to the magnetic field, and ends with the plane of the ring perpendicular to the magnetic field, what is the change in its magnetic flux? So, we start with this, let's just say that the magnetic field is initially going up, I mean, sorry, it's always going up, this is initial, this is final.

Initially the ring has it's plane parallel to the magnetic field, this is how the ring looks like, finally the ring has its plane perpendicular to the magnetic field. Now, the question is where is the normal? Right? And the normal it's going to be the crust answer in this because the magnetic flux is B, A, cosine of theta, we need to know what theta is, and theta is going to be what changes from the initial scenario to the final scenario, okay? We know what B is, the problem tells us enough to figure out what theta is, the only thing we're really have to calculate is the area, because we're told the radius, not the area. So itÕs a circle, right? It's a ring, this is going to be pi, r squared, which is pi 0.02 squared, you have to convert from centimeters to meters, don't forget, and so the area is going to be 0.0013 square meters. Now, we have all three pieces of information. Okay, initially, let's look at the picture on the left, when the plane is parallel to the magnetic field, the normal is perpendicular to the magnetic field, what is cosine of 90 degrees going to be? It's going to be 0, so the flux, initially is just 0. Now, looking at our final scenario, when the plane is perpendicular to the magnetic field lines, in this case the angle is 0 degrees, so the flux is going to be the magnetic field, right? We're told the 0.6 times, the area times cosine of 0 degrees, and what is cosine of 0 degrees? It is just 1, right? Or you can plug everything in your calculator and you would find that the answer is just 6.8 times 10 to the negative 4 Webers. So finally, our answer is, the change in magnetic flux is the final, right where I was standing, minus the initial to 0, so it's just the final answer or 6.8 times 10 to the negative 4 Webers. Okay guys, thanks for watching.

A circular coil with 150 loops and a radius of 10 cm is initially oriented in a 0.5 mT magnetic field, with the axis of the coil and the field at a 30^{o} angle. After some time, the coil rotates so that the field is at an angle of 40^{o} to the plane of the coil, and the field strength drops to 0.3 mT. What is the change in magnetic flux during this time?

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Consider a circular loop of radius R in the presence of a uniform magnetic field. Which of the following statements is true?

A) Doubling the magnetic field would result in the same change in flux as doubling the radius would.

B) Doubling the angle between the magnetic field and the surface will double the magnetic flux.

C) Changing the angle between the axis of the coil and the field from 90^{o} to some θ angle would result in the same change in flux as doubling the area of the coil.

D) Halving the radius of the coil halves the magnetic flux through the coil.

E) None of the above are true.

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A circular coil with 200 loops and a radius of 0.15 mm is placed 12 m from a straight, current-carrying wire as shown in the following figure. What is the magnetic flux through the coil? Assume that the magnetic field is roughly constant through the coil, since 12 m >> 0.15 mm.

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A circular coil with a 15 cm radius and 150 loops is oriented in the xy-plane. If a uniform 5 mT magnetic field is oriented 37^{o} above the xy-plane, what is the magnetic flux through the coil?

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A 2.0 m conductor is formed into a square and placed in the horizontal xy-plane. A magnetic field is oriented 30.0° above the horizontal with a strength of 3.0 T. What is the magnetic flux through the conductor?

A) 6.0 T • m^{2}

B) 0.65 T • m^{2}

C) 0.75 T • m^{2}

D) 0.37 T • m^{2}

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A 2.0 m conductor is formed into a square and placed in the horizontal xy-plane. A magnetic field is oriented 30.0° above the horizontal xy-plane with a strength of 1.0 T. What is the magnetic flux through the conductor?

A) 0.25 T•m^{2}

B) 0.12 T•m^{2}

C) 0.22 T•m^{2}

D) 2.0 T•m^{2}

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A conducting wire with a total length of 2.0 m is formed into a square and placed in the horizontal *xy*-plane. A magnetic field is oriented 30.0° above the horizontal with a strength of 10.0 T. What is the magnetic flux through the conductor?

A) 2.5 T • m^{2}

B) 20 T • m^{2}

C) 1.2 T • m^{2}

D) 2.2 T • m^{2}

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The trig. function of angle φ between vectors **B** and **A **in the magnetic flux should be (sine / cosine) __________

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