A 12-cm-diameter, 1.0-m-long solenoid is wound with 2000 turns...

Question

A 12-cm-diameter, 1.0-m-long solenoid is wound with 2000 turns of superconducting wire. When the magnet is turned on, the current increases from 0 to I_max in 2.5 s. At t = 1.0 s, the induced electric field midway between the axis and the windings is 7.5×10⁻³ V/m. What is the solenoid's steady magnetic field strength?

Accepted Answer

Welcome back. Everyone in this problem, consider a solenoid with a length of 2.2 m and a diameter of 15 centimeters. It is w with 2500 turns of a highly conductive wire and at an equidistant point from the solenoid's axis and its windings, the induced electric field after one second is 6.8 times 10 to the negative three volts per meter. If it only takes four seconds for the current within the solenoid to soar from zero to its peak value. When the magnet is activated, we want to determine the solenoids magnetic field when the current has stabilized. For our answer choices. A is 0.6 teslas B is 0.7 teslas C is 1.2 teslas and the D is 2.2 teslas. Now, what do we know here? Let's try to visualize what's going on. OK. So we have our cleo it probably looks something like this, right? And we know that it's 2.2 m long, we also know that it has a diameter of 15 centimeters. So if I were to draw that here, we know its diameter is 15 centimeters and we also know that it's wound with 2500 turns. So N equals 2500. What else do we know? We also know that at an equidistant point from the solenoids axis and its windings, the induced electric field after one second is 6.8 times 10 to the negative three volts per meter. So E equals 6.8 times 10 to the negative three volts per meter. And we want to figure out the solenoids magnetic field when the current has stabilized, which we know that it takes four seconds for the current within the sole to soar from zero to its peak value. So when it achieves its peak value, that's when the current has been stabilized. Again, we also know that it's a highly conductive wire. So there's zero resistance in the wire. So we want to find that the solenoids magnetic field which is b. So here's the question, how are we going to figure that out? Well, let's think about what's going on here. Do we know anything about or or changing mag in, in, in magnetic flux based on the information that we have? Well, we know that there's an induced EMF for changing flux. So recall that we have an induced electromotive force which requires a change in flux over time. Nor do we know anything else about an induced electromotive force? Well, we also know as it relates to our electric field, the induced, let me write it beside it, the induced EMF equals our electric field multiplied by two pi R. OK. So if we combine both of those, because both of those are equal to the induced EMF this tells us then that E multiplied by two pi R which is the length of our, the, the length of our uh circular coil equals and equals are changing flocks over time, right? So we, we know that so far. So that means then that we can even rearrange this to sulfur our electric field. But remember what do we want? We want the solenoids magnetic field. So is there any way to link the magnetic field to the information here? Well, recall recall that our change in flux equals or change in. Well, rather let me say that recall that flux, OK, flux equals the magnetic field multiplied by the area. So if that's true, our change in flux will be equal to the change in the magnetic field multiplied by the area. But remember our area will be constant. We're assuming a constant area here. So that means that will be equal to our changing magnetic field multiplied by a. So here we have a value for the change in magnetic field, we should be able to substitute it to solve for whatever that magnetic field will be eventually because no, this tells us that by substituting our value for the change in flux, E multiplied by two pi R is going to be equal to our changing magnetic field multiplied by a our area all divided by the change in time. Now we know that our area would be equal to pi R squared. So really is equal to pi R squared, those things stay constant multiplied by our change in magnetic field over time. So if we transpose here, we should be able to solve for a changing magnetic field. And if we can find that change in magnetic field over time, then we will be able to find a magnetic field after four seconds because no, this implies. So if we do some transposition here, we can divide both sides by pi R squared, which means the change in magnetic field over time. OK equals E multiplied by two pi R all divided by pi R squared. If we cancel pi and cancel uh R here from our radii, then this tells us that our changing magnetic field over time, right equals two multiplied by E divided by R where R is the radius of archive. Now this value, this value gives us our change per second. So if we're going, if we can figure out the change per second, we can multiply it by the time that has passed, which is four seconds to figure out the magnetic field when the current has stabilized. OK. So that means at four seconds. OK. Or let me say after right after four seconds right now, our magnetic field, the magnitude of our magnetic field divided by four, right equals two E divided by R which then tells us that the strength of our magnetic field will be four multiplied by two E which is eight E divided by R. Now, we're almost there. Now, we can find the magnetic field because we already know the electric field strength after one second and we know the radius of our choir. So substitute in that recall that our electric field strength is 6.8 times 10 to the negative three. So that's gonna be eight times 6.8 times 10 to the negative three volts per meter. And that will be divided by the radius of the coil, which if we remember its diameter is 15 centimeters, so its radius would be 7.5 centimeters. So that's 7.5 multiplied by 10 to the negative 2 m. No, when we work that out, we'll find that our magnetic field equals 0.7 teslas rounded to one decimal place. Therefore, our magnetic field is 0.7 teslas. When we look back in our answer choices that would have been answer. Choice B Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Magnetic Field in a Solenoid

Induced Electric Field

Time-Varying Current

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