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Hello everyone,


"The heating element of a toaster is a long wire of some metal, often the alloy nichrome, which heats up when a potential difference is applied across it. In the U.S.A., plugging a toaster into the wall outlet is equivalent to applying a 120-V potential source across it. For these problems consider a 300-W toaster connected to a wall outlet."


"How could one increase the rate at which heat is produced?"

A. Use a longer wire

B. Use a thicker wire

C. Both A and B

D. None of the above


Rate of heat produce brings to mind power, P=IV=I^{2}R=V^{2}/R,

but all these questions are related to each other by Ohm's law, so is it not equally valid to use either formula? The difficulty comes in when dealing with resistance because it is either directly or inversely related in different forms. 


R, for resistance = rho l/A, for rho being the resistivity, l being the length, and A being the area. 


So to increase power for heat generated, I could increase the amount of R through the increase of length supporting the I^{2}R equation, but contradicts the V^{2}/R.  Or I could increase power through the increase of thickness in the wire supporting V^{2}/R, but contradicting I^{2}R. 



The answer given was B.

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Given a constant voltage, the amount of power dissipation ( == heat ) in the element is proportional to current.  Current will increase as resistance falls.  Increasing the cross-section of the element reduces resistance (it's like having more resistors in parallel).


I suppose you could get a similar effect by reducing the length, but as it approaches a short circuit, the element, being less robust, is more inclined to fry.  As we Sparkies say:  "It's smoke that makes those things work.  If you let the smoke out, they don't work any more."

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