Question

The heat produced in a wire carrying an eletric current depends on the current, the resistance and the time.

Assuming that the dependence is of the product of powers type, guess an equation between these quantities

using dimensional analysis. The dimensional formula of resistance is $M{L}^2{I}^{-2}{T}^{-3}$and heat is a form of energy.

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Let the heat produced be H, the current through the wire be I, the resistance be R and the time be t.

Since heat is a form of energy, its dimensional formula is $M{L}^2{T}^{-2}$.

Let us assume that the required equation is

H = $k{I}^a{R}^b{t}^c$,

Where k is a dimensionless constant

Writing dimensions of both sides,

$M{L}^2{T}^{-2}=I^a(M{L}^2{I}^{-2}{T}^{-3}{)}^b{T}^c$

= $M^b{L}^{2b}{T}^{-3b+c}{I}^{a-2b}$

Equating the exponents,

b = 1

2b = 2

-3b + c = -2

a - 2b = 0

Solving these, we get, a = 2, b = 1 and c = 1.

Thus, the required equation is $H=k{l}^{2}Rt$