A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a r...

Question

A set of Helmholtz coils (see Problem 62, Fig. 28–61) have a radius 𝑅 = 10.0 cm and are separated by a distance 𝑅 = 10.0 cm . Each coil has 85 loops carrying a current I = 2.0 A. (a) Determine the total magnetic field B along the 𝓍 axis (the center line for the two coils) in steps of 0.2 cm from the center of one coil (𝓍 = 0) to the center of the other (𝓍 = 𝑅 ).

Accepted Answer

Hi, everyone. Let's take a look at this park's problem deal with magnetic fields. This problem says along with the Z axis, two identical Hempel coils, each with a radius of 12.0 centimeters and consisting of 200 turns of wire are placed at a distance of 12.0 centimeters from each other. A current of 3.0 amps flow through both coils in the same direction, determine the total magnitude of the magnetic field along the axis at a 0.6 0.0 centimeters from the center of the first coil. And we're given a hint that says at a distance Z from the center along its axis. The magnetic field due to a coil with N turns is B as a function of Z is equal to mu not nir squared divided by the quantity of two multiplying the quantity of R squared plus Z squared in quantity raised to the three halves. Below the question. We're given a diagram of what was described in the problem. We're also given four possible choices as our answers. For choice. A we have 0.0025 tesla. For choice B, we have 0.0038 tesla. For choice C, we have 0.0045 tesla. And for choice D, we have 0.0058 tesla. Now the question asks us to calculate the total magnetic field. And so our total magnetic field is gonna be made up of the magnetic fields from each of the coils. And we're given a, an equation for the magnetic field or a coil. So that's where we're gonna start. And so we'll calculate that magnetic field for each of the coils and then add them together to find the total magnetic field. So starting off with our first coil will label its magnetic field as B one and that's gonna be equal to you not in ir squared divided by the quantity of two multiplied by the quantity of R squared plus Z squared in quantity raised to the three halves. And we actually have values for all the variables on the right hand side. So we'll just go ahead and plug those in, we'll have B one equal to from U kno that's a constant that is, or pi multiplied by 10 to the negative seven tesla meters per amp. That's gonna be multiplied by the number of turns, which we were told in the problem is 200 then that's gonna be multiplied by the current, that's what I represents. And so that is three amps and then that gets multiplied by the radius which is R, we're given that the problem that is 12 centimeters. But I'm gonna need that in meter. So I'll divide by 100 and that will give me about uh the radius of 0.12 m. And that quantity is gonna be squared. Then all of that is divided by the quantity of two, multiplying the quantity of R squared. Again, we have an R value of 0.12 m and that's squared, then we'll have plus coordinate Z. We're given that in the problem as well. That's gonna be the six centimeters again, defined by 100 to convert that into meters will have 0.06 m. And that quantity is squared. And then that whole quantity is raised to the three halves. So we could plug those values into our calculator. I'll get B one is equal to 0.00 225 tesla. And here have just kept three significant figures. We're gonna repeat the same thing for the other coil. And so we'll call that B two and that's gonna be equal to the same formula which is Muno nir squared divided by the quantity of two multiplying the quantity of R squared plus Z squared in quanti raised to the three halves. And we're actually going to plug in the same values that we have for B one, we'll have B two is equal to, for Muno that is steel, four pi multiplied by 10 to the negative seven tesla meters per amp. This coil also had 200 turns. So multiplied army not by 200 that gets multiplied by our current, which is three amps again, and that's gonna be multiplied by the radius squared, which again is 12 centimeters. So that's gonna be 0.12 m. That quantity squared that could be divided by the quantity of two multiplying the quantity of R squared which is 0.12 m squared plus Z. And since it's the midpoint, we're again +66 centimeters from the second coil, so it's gonna be 0.06 m, it's gonna be squared and that whole quantity is raised to three halves. So we'll get B two is equal to 0.00225 tesla. So now to find the total magnitude, we just need to add these two magnitudes together. And that's because the magnetic fields from each of the coil are pointing in the same direction. And in this case, as shown in the diagram, these um magnetic fields will be pointing downward. And so we'll get our to magnetic field B gonna be equal to B one plus B two. And so B is going to be equal to 0.00225 tesla plus 0.00225 tesla. And so that means that B gonna be equal to 0.0045 tesla. And here we just kept two significant figures. And if I look at my answer choices, this corresponds to answer choice C so just a quick little recap of what we've done, we started off by calculating the magnetic field for each of the coils, given the equation um that we were given in the problem. And then once we had the two magnetic fields, we simply added their magnitudes together because our magnetic fields were pointing in the same direction in this case downward. And so once we added the two magnitudes together that gave us our total magnetic field, so I hope that this has been useful and I'll see you in the next video.

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