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.