A solenoid 25.0 cm long and with a cross-sectional area of 0.500 cm^2 contains 400 turns of wire and carries a current of 80.0 A. Calculate: (c) the total energy contained in the coil's magnetic field (assume the field is uniform);

Video transcript

Hey, everyone. So this problem is dealing with electrical energy. Let's see what it's asking us. A 300 turn solenoid is used in a device that has a length of 15 centimeters and a cross sectional area of 150. centimeters squared. The wire supports a maximum current of 60 amps. And we're asked to determine the energy stored in that uniform magnetic field of the solenoid when it has maximum current. Our multiple choice answers here are a 0.18 jules b 18 jules, C 0.27 jules or D 2.72 jules. So the key to this problem is recalling that energy stored in a uniform magnetic field at a maximum current as a special equation. It's given by U capital U is equal to lower case U or our energy density multiplied by V our solenoid volume. So we can solve for both lower case U our energy density and our solenoid volume to finally get to this um new maximum energy. So first, we can recall that U is given by or you are lower case U excuse me, our energy density is given by B squared divided by Muno. And in turn B, our magnetic field is given by and five divided by L. And so we can solve for B and then we can solve for our energy density. So recall that you not is the permittivity of free space that is a constant four pi times 10 to the negative seven. And that unit is tesla meters per A N is the number of terms. So that was 300 then multiplied by our maximum current 60 camps, all of that divided by the length of the wire. The problem it was given to us is 15 centimeters. We need to remember to keep everything in standard units. So I'm going to rewrite that as 150.15 m and and we get a um magnetic field of 0. teslas. Now we can go back up to our energy density equation and plug that in. So we have 0.151 tesla squared divided by, alright, I'm sorry, there's a, this should be um B squared divided by two. You not. So it's gonna be two multiplied by four pi times 10 to the negative seven tesla meter or amp. And that gives us here a energy density uh three times oops, sorry, sorry. N 9072 jewels for meter, cubed, 9072 jewels for me are cued. Ok. So now we have solved for our energy density. And the next step is solving for our. So probably so our solenoid volume is simply going to be the length of the solenoid multiplied by the cross sectional area. And so the length was given again as 0. m. And then our cross sectional area, when we take it from centimeters squared to meter squared, it's going to be two times 10 to the minus five m squared. And so that gives us the volume of three times 10 to the negative six m cubed. So then our last step is to just plug those found values in for our energy. So that looks like energy is equal to Jews per meter cubed multiplied by three times 10 to the negative six m cubed. And that comes out to 0.27 Jews. And so that is the final answer for this problem. And when we look at our multiple choice solutions, it aligns with answer choice C so that's all we have for this one, but we'll see you in the next video.