Conductors by Electric Currents. 267 



and therefore the surface of maximum temperature is the 

 equipotential V'-iCV.+V,), 



and the maximum temperature is 



or 



jcW/Sk, 



where E is the applied electromotive force. 



From the examples considered in the following section 

 a fair estimate of the time taken in establishing the steady 

 state in this general case may be derived, and also of the 

 disturbing effect produced by emission of heat from the 

 bounding surface, which may be useful when it is difficult to 

 obtain complete solutions by the methods of Section 3. 



5. Examples of the Varying Stale. 



When, as in the preceding section, the solution for the 

 steady state can be found in the form 



k$= A + BV-± jcV 2 , 



it will be convenient, in applying the method of Section 3 

 to find the solution for the intermediate varying state, to 

 assume 



U = ^ + iicY 2 -(A + BY), 



so that the final state will be given by U = 0. U will still be 

 determined by (F), and the initial condition will be 



U = i/cV 2 -(A + BV), 



if initially 3 = throughout. 



As an example, take the case of a cylindric bar of length 

 2/, initially at temperature zero throughout, having its ends 

 maintained at potentials + V , and at temperature zero, and 

 its surface being impenetrable to heat. 



The solution for the steady state is by last section, 



the origin being taken at the middle of the bar. Now, noting 

 that the ends are to be maintained at temperature zero, and 

 expanding suitably by Fourier's Theorem, we get 



.■■Wj *Vi.-. V| ,82» (-)' C0S (2n + lW 



