Question

A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).​

Answer 1

Answer:

he Laplace equation is

Let u = X(x) . Y(y) be the solution of (1), where „X‟ is a function of „x‟ alone and „Y‟ is  a function of „y‟ alone.

Now the left side of (2) is a function of „x‟ alone and the right side is a function of „t‟ alone. Since „x‟ and „t‟ are independent variables, (2) can be true only if each side is equal to a constant.

 

Solving equations (3), we get

 

 (i) when „k‟ is positive and k = l2, say

 

X = c1 elx  + c2  e - lx

Y = c3 cosly + c4 sin ly

 

 

(iii)               when   „k‟   is   zero.

X = c9  x + c10

Y = c11 x + c12

 

Thus the various possible solutions of (1) are

Of these three solutions, we have to choose that solution which suits the physical nature of the problem and the given boundary conditions.

Step-by-step explanation:

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Answer 2

A rectangular plate with insulated surface is 10 cm. wide and so long compared to its width that it may be considered infinite length. If the temperature along short edge y = 0 is given u(x,0) = 8 sin(px/ 10) when 0 <x <10, while the two long edges x = 0 and x = 10 as well as the other short edge are kept at 0o C, find the steady state temperature distribution u(x,y).