CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0...

Question

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Find an expression for the induced current I(t) as a function of time.

Accepted Answer

Hey, everyone. Let's go through this practice problem. A solenoid with 15 turns that has a resistive value of R equals 10 is placed within a time varying magnetic field. B of 0.03 T plus 0.015 T squared at teslas. This external magnetic field is oriented perpendicular to the plane of the solenoid. Derive an expression for the current I of T that is induced in the solenoid due to this magnetic field as a function of time, the radius of the solenoid is two centimeters. Option A 5.7 multiplied by 10 to the power of negative four, all multiplied by one plus T api. Option B 4.8 multiplied by 10 to the power of negative three, all multiplied by one plus T appears option C 6.3 and multiplied by 10 to the power of negative four. And then all multiplied by the same thing. And option D 8.1 multiplied by 10 to the power of negative three and all multiplied by the same thing. So because we have a magnetic field B that is varying with respect at time, that means Faraday's law is going to be relevant because the current is being induced as a result of that magnetic field changing. So to start us off, let's recall what fda's law says. It says that the EMF of an induced current is equal to the number of turns within the coil multiplied by the rate of change of the magnetic lox or expanding this out a bit. The magnetic flux can also be written as the value of the magnetic field multiplied by the surface area of the coil of wire through which we're inducing the current. So for our purposes, the surface area is not changing. So we can pull the eight out of that derivative. So N a multiplied by the rate of change of the magnetic field. And since the solenoid is described as being circular, since that's what a solenoid is, we can substitute in for a the surface area for the, the, the formula for the surface area of a circle which recall that it is pi multiplied by the square of the radius. So now we've expanded our formula out quite a bit. Our formula is now mostly in terms of things that the problem gives us. But the rate of change of the magnetic field might still lead to some confusion. So let's talk about that separately for a bit because we are given a formula for the magnetic field with respect to time. But we need to take a derivative of it. It needs to be with respect to time in order for it to work in this equation. So let's actually look at the formula we've been given, we're told that the magnetic field is 0.03 T plus 0.015 T squared. So in order for us to put this into our Frida's Law formula, we'll need to take the derivative. And it's a pretty simple derivative. This is a pretty basic power rule derivative. So for the first term 0.03 T, the T goes away as per the power rule. So we just have a 0.03 then plus 0.015 T squared. Once again, applying the power rule, we take the exponent two and multiply it by the coefficient 0.015. So we're left with 0.03 in our coefficient multiplied by T. And as per the power rule, we don't write out the square because we subtract one. So it's basically an exponent of one. And since we have a 0.03 as both coefficients, we can simplify this a bit by factoring out the 0.03. So we have 0.03 multiplied by in parentheses, one plus T. So let's put that into our Faraday law formula and start plugging in numbers. So as per the formula we discussed earlier using Faraday's law, this is equal to N the number of turns, which is given to us in the problem as 15 turns of the solenoid multiplied by pi multiplied by the square of the radius of the solenoid. So that would be two centimeters given to us in the problem. So 0.02 m that's being squared and then multiplied by the rate of change of the magnetic field. So that's 0.03 multiplied by one plus T. Now, obviously, we can't really do any math with the one plus T since we don't know what T is, it is a, it is a dependent variable in this case. So it can always be different. But ignoring that and putting everything else into our calculator, we find a new coefficient of about 5. multiplied by 10, raised to the power of negative four, multiplied by in parentheses, one plus T. So this is our EMF for the induced current. But the problem is asking us to find a formula for the current itself and this is where we're going to have to bring S law into it because we're given the resistance of the solenoid. So recall that according to S law, the current within a circuit is equal to the potential difference or the voltage or the EMF divided by the resistance. But this problem tells us that the resistance has a value of one. So, so this is some pretty easy math to do. We'll just have to take the EMF we found in volts and just simply divide it by 10 which is essentially just not going to change anything other than the units as we divide this by 10. And we of course, just end up with the exact same value but with the current units of API, so 5.7 multiplied by 10 to the power of negative four, multiplied by one plus T API. And that is our answer to this problem. And if we look at our multiple choice options, we can see that this agrees with option A. So option A is the correct answer to this problem. And that is all for this video. I hope this video helped you out and please consider checking out some of our other tutoring videos which will give you more experience with these types of problems, but that's all for now. And I hope you all have a lovely day. Bye bye.

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Find an expression for the induced current I(t) as a function of time.

Key Concepts

Faraday's Law of Electromagnetic Induction

Ohm's Law

Magnetic Flux

Watch next

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t2, where B is in tesla and t is in seconds. Find an expression for the induced current I(t) as a function of time.

Key Concepts

Faraday's Law of Electromagnetic Induction

Ohm's Law

Magnetic Flux

Watch next

CALC A 5.0-cm-diameter coil has 20 turns and a resistance of 0.50 Ω. A magnetic field perpendicular to the coil is B = 0.020t + 0.010t², where B is in tesla and t is in seconds. Find an expression for the induced current I(t) as a function of time.