The magnetic field B at all points within the colored circle show...

Question

The magnetic field B at all points within the colored circle shown in Fig. E29.15 has an initial magnitude of 0.750 T.

(The circle could represent approximately the space inside a long, thin solenoid.) The magnetic field is directed into the plane of the diagram and is decreasing at the rate of -0.0350 T/s. (c) What is the current in the ring if its resistance is 4.00 Ω?

Accepted Answer

Hi, everyone. In this particular example, we are asked to actually find the current going through a circular loop with a radius of 20 centimeters inserted into a magnetic field directed into the page where the magnetic field will initially have the strength or magnitude of 0.85 tesla and a decreasing rate of minus 0.4 tesla per seconds. We will have the resistance of the ring being at five a.m. And we were asked to find the current going through the system. So first, we want to actually just list out all the things that is given in this particular example. First, we have the radius, which is the R R is going to be 20 centimeters, which is going to just be 0.2 m. And then we will have the magnitude of the initial magnetic field which is 0.85 Tesla. And we have the rate of decrease, which is the DB over DT which will equals to -0.04 tesla per second. And lastly, we have the resistance which is at five Pope. Okay. So the way you wanna sulfur this example is to just by using Faraday's law, we were asked to actually find the current. And the way we can relate the current with our days law is to by finding the induced CMF, we call that the current will equals to induce CMF over our. So we first have to find the INDIE CMF and doing that, we can use Faraday's law which will give us the induced CMF by actually finding the absolute value of the D or the change in the magnetic flux over D T just like. So okay. So we want to find the magnitude of the INDIE CMF by looking into the value of the change in magnetic flux over time. So we want to first recall again that the magnetic flux can be calculated by multiplying B and A together. And for this, the A is going to essentially just be the area of a circle. So this is just going to be be multiplied by pi R squared. And we can substitute defy be into this formula right here so that it will give us over DT. And we know that the pi R squared is going to be constant. So we can just pull that out and we will get the defi B over D T value of pi R squared multiplied by D B over D T just like. So, so we can essentially just calculate the induced E M F by multiplying pi R squared with D B over D T which is exactly what we're going to do in this particular example. Okay. So the induced E M F here is going to just be the D five B over DT, which is going to equals two pi R squared multiplied by D B over D T. And because we're looking at the only the magnitude of the induced CMF we can kind of absolute value the D B over D T S. Well, so plugging all this value, we get 0.2 m squared multiplied by 0.4 tesla per second. And then we will get an induced E M F of five point oh three times 10 to the power of minus three fault just like. So, so now we can actually calculate the, the induced current that we have. The current really goes to the Indy CMF over the resistance and we have the E M F 3503 times 10 to the power of minus three fold and the resistance to be five. And this will give us a value of 1.01 times 10 to the power of -3 amp for the current. And that will correspond to option deed that we have here. And that will essentially be it for this particular problem. If you guys have any sort of confusion, please make sure to check out our other similar topics, lesson videos and that will be all for this particular example. Thank you.

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