The flux of magnetic field through a closed conducting loop changes with time according to the equation, Φ = at^{2} + 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.

#### Solution

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 = at^{2} + bt + c

(a) The dimensions of the quantities at^{2}, bt, c and ϕ must be the same.

Thus, the units of the quantities are as follows:-

\[a = \left( \frac{\phi}{t^2} \right) = \left[ \frac{\phi/t}{t} \right] = \frac{Volt}{s}\]

\[b = \left[ \frac{\phi}{t} \right] = \text{Volt}\]

\[c = \left[ \phi \right] = \text{Weber}\]

(b) The emf is written as:-

\[E = \frac{d\phi}{dt}=2at+b=2\times0.2\times2+0.4..........\left(\because a=0.2, b=0.4\text{ and }c=0.6\right)\]

On substituting t = 2 s, we get

E = 0.8 + 0.4

= 1.2 V