I Electricity is distributed from electrical substations to neighborhoods at 15,000 V. This is a $60\mathrm{Hz}$oscillating $\left(\mathrm{AC}\right)$ voltage. Neighborhood transformers, seen on utility poles, step this voltage down to the $120\mathrm{V}$that is delivered to your house.

a. How many turns does the primary coil on the transformer have if the secondary coil has 100 turns?

b. No energy is lost in an ideal transformer, so the output power $P_{\text{out}}$from the secondary coil equals the input power $P_{\text{in}}$ to the primary coil. Suppose a neighborhood transformer delivers $250\mathrm{A}$ at$120\mathrm{V}$. What is the current in the $15,000\mathrm{V}$ line from the substation?

(a) In order to step down the voltage, the primary coil must have more coils than the secondary coil. Faraday's law states that the number of turns N is directly proportional to the induced emf. We combine the number of turns and the induced emf for the two coils using equation (30.30)in the form

$\frac{V_2}{V_1}=\frac{N_2}{N_1}$

Where$V_1$and $N_1$are the emf and the number of turns for the primary coil while $V_2$and $N_2$are the emf and the number of turns for the secondary coil. We rearrange equation (l) for $N_1$to be in the form

$N_1=\frac{V_1}{V_2}{N}_2$

Now, we plug the values for $V_1,N_2$and $V_2$into equation (2) to get $N_1$

$N_1=\frac{V_1}{V_2}{N}_2=\left(\frac{15000\mathrm{V}}{120\mathrm{V}}\right)\times 100\text{turns}=12500\text{turns}$

(b) When a current flows through a resistor, it dissipates energy; the rate of energy dissipation is the power. The process by which energy is transferred from the current to the resistor is

$P=IV$

The output power is at the secondary coil where $I_2=250\mathrm{A}$and the voltage is $V_2=120\mathrm{V}$. So, we get the output power by

$P_2=I_2{V}_2=\left(250\mathrm{A}\right)\left(120\mathrm{V}\right)=3\times {10}^4\mathrm{W}$

Let us use equation

(3) to get$I_1$

$I_1=\frac{P_1}{V_1}=\frac{3\times {10}^4\mathrm{W}}{15000\mathrm{V}}=2\mathrm{A}$

(a) $N_1=12500$turns.

(b)$P_1=2\mathrm{A}$.