Question

The flux of magnetic field through a closed conducting loop changes with time according to the equation, Φ = at² + bt + c. (a) Write the SI units of a, b and c. (b) If the magnitudes of a, b and c are 0.20, 0.40 and 0.60 respectively, find the induced emf at t = 2 s.

Answer 1

According to the principle of homogeneity of dimensions, the dimensions of each term on both the sides of a correct equation must be the same.
Now,
ϕ = at² + bt + c
(a) The dimensions of the quantities at², bt, c and ϕ must be the same.
Thus, the units of the quantities are as follows:
$$\begin{gathered} a=\left(\frac{\mathrm{\varphi }}{t^2}\right)=\left[\frac{\mathrm{\varphi }/t}{t}\right]=\frac{\mathrm{Volt}}{\mathrm{s}} \\ b=\left[\frac{\mathrm{\varphi }}{t}\right]=\mathrm{Volt} \\ c=\left[\mathrm{\varphi }\right]=\mathrm{Weber} \end{gathered}$$

(b) The emf is written as:
$E=\frac{d\varphi }{dt}$ = 2at + b = 2 × 0.2 × 2 + 0.4 (∵ a = 0.2, b = 0.4 and c = 0.6)
On substituting t = 2 s, we get
E = 0.8 + 0.4
= 1.2 V