By considering a wire of radius 'r', lenght 'l' and resistivity 'e' through which a current I flows, show that
a) the rate of production of heat by it is I^2*P*l / pi*r^2
b) the rate of loss of heat from its surface is 2pi*r*t*h , where 'h' is the heat lost per unit area of surface per second.
To derive the given expressions, let's break down the problem into two parts:
a) Rate of production of heat by the wire.
The rate of production of heat is given by the formula:
Power = I^2 * R
Where R is the resistance of the wire. The resistance of a wire is given by:
R = (ρ * l) / (A)
Where ρ is the resistivity of the material, l is the length of the wire, and A is the cross-sectional area of the wire.
The cross-sectional area of a wire with radius r is given by:
A = π * r^2
Substituting the value of A in the resistance equation, we get:
R = (ρ * l) / (π * r^2)
Therefore, the power produced by the wire can be written as:
Power = I^2 * R
= I^2 * [(ρ * l) / (π * r^2)]
= (I^2 * ρ * l) / (π * r^2)
This expression shows that the rate of production of heat by the wire is I^2 * Power * l / (π * r^2).
b) Rate of loss of heat from the wire's surface.
The rate of loss of heat from the wire's surface can be calculated using the formula:
Heat lost = Area * Heat lost per unit area per second
The surface area of the wire is given by:
Surface area = π * r * l
Therefore, the rate of loss of heat from the wire's surface can be written as:
Rate of heat loss = Surface area * Heat lost per unit area per second
= (π * r * l) * h
This expression shows that the rate of loss of heat from the wire's surface is 2π * r * l * h.
Overall, we have derived the expressions for both the rate of production of heat by the wire and the rate of loss of heat from its surface.