Question

Prove that energy is conserved during electromagnetic induction.

Answer 1

The mechanical energy of the spring-mass system is given by

E = `1/2 "mv"^2 + 1/2 "kx"^2`     ......(1)

The energy E remains constant for varying values of x and v. Differentiating E with respect to time, we get

`"dE"/"dt" = 1/2 "m"("2v" "dv"/"dt") + 1/2"k" (2"x" "dx"/"dt") = 0`

or `"m" ("d"^2"x")/"dt"^2` + kx = 0    ......(2)

Since `"dx"/"dt" = "v" and "dv"/"dt" = ("d"^2"x")/"dt"^2`

This is the differential equation of the oscillations of the spring-mass system. The general solution of equation (2) is of the form
x(t) = X_m cos (ωt + φ) …… (3)
where X_m is the maximum value of x(t), ω, the angular frequency and φ, the phase constant. Similarly, the electromagnetic energy of the LC system is given by

U = `1/2 "Li"^2 + 1/2 (1/"C") "q"^2` = constant   ....(4)

Differentiating U with respect to time, we get

`"dU"/"dt" = 1/2"L" ("2i" "di"/"dt") + 1/"C" ("2q" "dq"/"dt")` = 0

or `"L" ("d"^2"q")/"dt"^2 + 1/"C" "q" = 0`    .....(5)

since i = `"dq"/"dt" and "di"/"dt" = ("d"^2"q")/"dt"^2`     .....(5)

Equation (2) and (5) are proved the energy of electromagnetic induction is conserved.