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Problem 60 Hard Difficulty

Imagine that a steady current $/$ flows in a straight cylindrical
wire of radius $R_{0}$ and resistivity $\rho$ . $(a)$ If the current is then
changed at a rate $d I / d t$ , show that a displacement current $I_{\mathrm{D}}$
exists in the wire of magnitude $\epsilon_{0} \rho(d I / d t) .$ (b) If the current
in a copper wire is changed at the rate of 1.0 $\mathrm{A} / \mathrm{ms}$ , deter-
mine the magnitude of $I_{\mathrm{D}}$ (c) Determine the magnitude of
the magnetic field $B_{\mathrm{D}}$ (T) created by $I_{\mathrm{D}}$ at the surface of a
copper wire with $R_{0}=1.0 \mathrm{mm}$ . Compare (as a ratio) $B_{\mathrm{D}}$
with the field created at the surface of the wire by a steady
current of 1.0 $\mathrm{A}$ .

Answer

a. $$\varepsilon_{0} \rho \frac{\mathrm{d} \mathrm{I}}{\mathrm{dt}}$$
b. $$1.5 \times 10^{-16} \mathrm{A}$$
c. $$1.5 \times 10^{-16}$$

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Video Transcript

All right. So this question we have to find out first place the expression for the displacement current. So from the deafness, um, off displacement current. What you can write is displacement current. That is I d. Is it called Tau? Absolutely. Times the chains in electric plugs with respect to time. So do you find E by DT? Right now, the changing electric flux is equal to what lick field into area Because they re created the flats portage area so into a Where is Constance alleviating that out on? That would be some of times, eh? D e by DJ Now, the electric field is the grand off potential. So what? I can varieties have still not 80 times. Oh, d I sorry my l DVD by DJ Because he's going to be If I am now this Internet cost to absolutely John absolute a bi l Now Weise I into our over our is constant. There is resistance. So I will have the eye by DT now using the equation for the resistance that say that resistance is receptivity. Terms lend upon the cross section So this gives me are a by l is equal to roll so just substitute bag This better Here, this gives me are a biological rule gives me rule. Absolutely not d ie vied ity. Is that what we wanted To show that the displacement current is the perfect off a rule So not And the rate of change up current We are done with a party of pushing Let's move on to part B my Given that the rate of change of current is deok tragedy is given embodies one impure for millisecond So that is, uh, housing in Pierre for a second. I'm so displaying the values. So to find our i d I read you could do now in the receptivity off a copper wire is if you can see the index are no way If you can find the copper wire has arisen. Studio 1.68 entered into the power minus eight on Absolutely not. We all know that if somebody is 8.85 entered into a minus toe Andi, I bet it is doing the question that is thousands. So this gives me i d is equal to one point for 8 to 10 to the power minus 16 and yours. All right, So we have found out the displacement current Now in the next part. See, I need to find the manager off the manicure. So B d is equal to what magnetic fit, you know, that is given by new, not the current. That is now. Since we need to find the displacement man, will you just listen, kind So use distressing current here. I want to buy now. Just look in the values here. So that is by inter 10 to the power minus seven. That is mu not the displacement. Current is one point for eight entered into the follow minus 16 to buy. So what is our are is given as one of millimeters sized into the power minus three meters. This gives me 24974 to 10 to the power minus 20 just now. All right, so we have also found out the displacement That man in different displacement current on the surface. Now let's compared this week, Field created the sufferance of the wire by steady current. So b d upon leaders call that B s the story. Current is equality Now. What is beauty? That is Drew client. Sorry. I think we need us off your special for this. So let me move this thing here from from here from here. Do you hear? So beauty by the scene is equal, Do you not? I d bye do by our now this divided by four b s that really knew Not I s by do by our So we are left with I d my I s right So I d He's given as one point for eight entered into the power minus 16 Onda We will know that this recording is one in here. So one that gives me the race You off. That's that's been my big trip to the city manager Failed. He's minus what before to about minus 16. So this ends decision to this problem.

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