The heat produced in a wire carrying an electric current depends on the current, the resistance and the time. Assuming that the dependance is of the product of powers type, guess an eqn. between these quantities using dimensional analysis. The dimensional formula of resistance is `ML^2 A^(-2) T^(-3)` and heat is a form of energy.

Explanation

Let the heat produced be H, the current through the wire be I, the resistnce be R and the time be t. Since heat is a form of energy, its dimensioN/Al formula is `ML^3T^-2`.
Let us assume that the required equation is
`h=kI^aR^bt^c`,
whre k is a dimensionless constant.
Writing dimension of both sides,
`ML^2T^_2=I^a(ML^2I^-2T^-3)^bT^c`
` =M^bL^(2b)T^(-3b+c)I^(a-2b)`
Equating the exponents,
`b=1`
`2b=2`
` -3b+c=0`
` a-2b=0` Soving these, we get `a=2, b1 and c=1`.
` Thus, the required equation is `H=kl^2 Rt.`