Part of a long, thin insulated straight wire is formed into a ...

Question

Part of a long, thin insulated straight wire is formed into a single circular loop of radius 𝑅 (Fig. 28–68) and carries a current I. (a) What is the magnitude and direction of the magnetic field at the center of the loop? (b) If the plane of the loop is twisted 90 degrees so that the plane is perpendicular to the straight part of the wire (i.e., in the yz plane) what is the magnitude and direction of the field now at the center of the loop?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A long insulated wire is shaped into a single circular loop with a radius R that carries a steady current eye in the clockwise direction. As shown in the figure, a magnetic sensor is placed at the center of the loop to measure the magnetic field. I, if the sensor measures only the field created by the loop, find the magnitude and direction of the magnetic field at the center I I, the wire is then twisted 90 degrees around the X axis so that the loop lies in the YZ plane with the current in the loop. Now flowing clockwise when viewed from the positive X axis, determine the magnitude and direction of the magnetic field at the sensor's position. Awesome. So it appears for this particular problem we're asked to solve for two separate answers. For part I, we're trying to figure out if the sensor is moved only. So if the sensor measures as we should focus for part I, so if the sensor measures only the field created by the loop, we're asked to figure out the magnitude and direction of the magnetic field at the center. That's what we're solving for. For part I, that's our goal. And then for I I, we're trying to figure out the magnitude and direction of the magnetic field at the sensor's position provided the conditions set. In part I I Awesome. So with that in mind, we know we now know that we're ultimately solving for two separate answers. So before we read off our multiple choice answers, to see what our final answer pair might be, let's quickly look at the figure that's provided to us. So as you could see, we have in the center of our figure, we have a circle with the center dot representing our magnetic center. So the magnetic sensor to the outer edge of the circle is some radius R and the current eye is moving downwards and going upwards and it's going in a clockwise direction. And as you could see, we have our Y and X AI labeled off to the side and we have our current eye at the very bottom of our circle and the current eye is moving from the right to the left. So it's moving to the left or our current. Awesome. So now that we have a visualization of what's going on in this particular prom, let's read off multiple choice answers, note that I'm gonna read for part I first and I'm gonna read for I, I, and I'll say that B equals some expression and then it's gonna be either into the plane or out of the plane and it's either gonna be along the positive X axis or along the negative X axis. So I'm just gonna say into plane or out of plane and I'm gonna say positive or negative. So A is two multiplied by pi multiplied by 10 to the power of negative seven, I divided by R into plane and two multiplied by pi multiplied by 10 to the power of negative seven I divided by R positive B is two multiplied by pi multiplied by 10 to the power negative seven I divided by R out of plane and two multiplied by pi multiplied by 10. The power negative seven I divided by R negative C is two multiplied by pi multiplied by 10 to the power negative seven I divided by R into plane and two multiplied by pi multiplied by 10 to the power negative seven I divided by R negative. And finally, D is two multiplied by pi multiplied by 10 to the power negative seven I divided by R out of plane and two multiplied by pi multiplied by 10 to the power negative seven I divided by R positive. OK. So now our next step is we need to recall a note that we're trying to focus on the f first we need to focus on the magnetic field at the center of the loop. So we're focusing on the loop only. So as we should recall a note for a circular loop with some current I, the magnetic field B at the center of the loop will be. And let us recall that this equation states that B is equal to mu zero where mu zero is the permeability of free space multiplied by I, the current all divided by two multiplied by R where R is the radius. So let's take a moment here to quickly write that Mu zero, which once again is our permeability of free space. And mu zero, the numerical value for that is equal to four multiplied by I so four multiplied by pi. And then we need to multiply pi by 10 to the power of negative seven. So mu zero probability of free spaces four multiplied by pi multiplied by 10 to the power of negative seven. And note that its units of measurement are teslas multiplied by meters per amp fantastic. So now moving right along here and once again, our, and our problem is the radius of the loop. So now all we need to do is plug in all of our known variables and solve for the numerical value of B. So we know the value for the probability of free space and we know that the current I which will just leave as I. So when we plug in our known variables, we'll get four multiplied by pi which up above, I used a multiplication symbol, but we could just use a simple dot instead of a cross symbol. So when you take four multiplied by pi multiplied by 10 to the power of negative seven, and to keep things simple, you don't really have to necessarily write the units of measurement at this particular moment. So we're just gonna leave them out for now. And then we can multiply it by R by I. And then we need to divide everything by two multiplied by R. So when we simplify, this will be all equal to two multiplied by pi multiplied by 10 to the power of negative seven I. And we need to divide everything by R fantastic. And that's when we simplified our answer. And that is our first answer for part I hooray, we did it, we've solved for our first answer. So now we can start solving for our second answer, which before we do that, we need to figure out the direction. So as we should recall and note that in order for us to solve for the direction we need to recall and use the right hand rule. So when we curl the fingers of our right hand in the direction of the current, which is clockwise, our thumb should point into the plane. So that means our second answer is into plane. So to keep things simple, I'm just gonna denote it as IP. So we're two planes. So this is IP so IP into plane, that's our first answer pair. So now we can start solving for I I. So now solving for our second and final answer, which is the magnetic field after a 90 degrees rotation. So this is the loop in the YZ plane. So as we should recall a note, the loop lies in the YZ plane with the current flowing clockwise when viewed from the positive X axis. So therefore, as we should recall a note, the field due to the loop in the YZ plane, this should be B equals two, multiplied by pi multiplied by 10 to the power of negative seven I divided by R. And that is our final answer. For part I I hooray, we did it. So now we need to figure out the direction. So once again, we're calling and using the right hand rule. So with our fingers curled in the clockwise direction, current direction. So this is from the positive xxe view, the thumb will point along the negative. So negative X axis. So I'm just gonna write negative fantastic. And that's it. That is our final pair of answers. Hooray, we did it. And this corresponds with multiple choice answer letter C which states that IB equals two multiplied by pi multiplied by 10 to the power of negative seven I divided by R into the plane. And for I I B equals two multiplied by pi multiplied by 10 to the power negative seven I divided by R along the negative X axis. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Magnetic Field Due to a Current Loop

Right-Hand Rule

Effect of Loop Orientation on Magnetic Field

Watch next