Chapter 21, Problem 77QAP

To determine

(a)

The power a 5.00 W electric razor will draw in Rome.

Answer to Problem 77QAP

The electric razor will draw a power of 20 W in Rome.

Explanation of Solution

Given:

  $\begin{array}{c}{V}_{\text{rms,Rome}}=240\text{ V}\\ f=60\text{ Hz}\\ P_{\text{USA}}=5.00\text{ W}\\ V_{\text{rms,USA}}=120\text{ V}\end{array}$

Formula used:

If the resistance of the razor is R, then,

  $P_{\text{USA}}=\frac{V_{\text{rms,USA}}^2}{R}$ and

  $P_{\text{Rome}}=\frac{V_{\text{rms,Rome}}^2}{R}$

Since the resistance remains the same,

  $R=\frac{V_{\text{rms,USA}}^2}{P_{\text{USA}}}=\frac{V_{\text{rms,Rome}}^2}{P_{\text{Rome}}}$

Therefore,

  $P_{\text{Rome}}=\left(\frac{V_{\text{rms,Rome}}^2}{V_{\text{rms,USA}}^2}\right)\left(P_{\text{USA}}\right)$......(1)

Calculation:

Substitute the known values of variables in equation (1) and calculate the power the razor draws in Rome.

  $\begin{array}{c}{P}_{\text{Rome}}=\left(\frac{V_{\text{rms,Rome}}^2}{V_{\text{rms,USA}}^2}\right)\left(P_{\text{USA}}\right)\\ ={\left(\frac{240\text{ V}}{120\text{ V}}\right)}^2\left(5.00\text{ W}\right)\\ =20\text{ W}\end{array}$

Conclusion:

Thus the electric razor will draw a power of 20 W in Rome.

To determine

(b)

The rms current that will run through the razor in United states and in Rome and also determine if it would damage the Razor in Rome.

Answer to Problem 77QAP

The rms currents that will run through the razor in United states is $4.17\times {10}^{-2}\text{A}$and in Rome is $8.33\times {10}^{-2}\text{A}$. Yes, it will damage the razor since the current in Rome is nearly 2 times that in USA.

Explanation of Solution

Given:

  $\begin{array}{c}{V}_{\text{rms,Rome}}=240\text{ V}\\ P_{\text{USA}}=5.00\text{ W}\\ P_{\text{Rome}}=20\text{ W}\\ V_{\text{rms,USA}}=120\text{ V}\end{array}$

Formula used:

The rms currents flowing in the razor in USA and in Rome are given by the expressions,

  $i_{\text{rms,USA}}=\frac{P_{\text{USA}}}{V_{\text{rms,USA}}}$......(2)

  $i_{\text{rms,Rome}}=\frac{P_{\text{Rome}}}{V_{\text{rms,Rome}}}$......(3)

Calculation:

Substitute the values of the variables in equation (2) and determine the value of $i_{\text{rms,USA}}$.

  $\begin{array}{c}{i}_{\text{rms,USA}}=\frac{P_{\text{USA}}}{V_{\text{rms,USA}}}\\ =\frac{5.00\text{ W}}{120\text{ V}}\\ =4.17\times {10}^{-2}\text{A}\end{array}$

Substitute the values of the variables in equation (3) and determine the value of $i_{\text{rms,Rome}}$.

  $\begin{array}{c}{i}_{\text{rms,Rome}}=\frac{P_{\text{Rome}}}{V_{\text{rms,Rome}}}\\ =\frac{20\text{ W}}{240\text{ V}}\\ =8.33\times {10}^{-2}\text{A}\end{array}$

The current in Rome is almost twice that of the current that flows in USA. Hence the razor would get damaged.

Conclusion:

Thus the rms currents that will run through the razor in United states is $4.17\times {10}^{-2}\text{A}$and in Rome is $8.33\times {10}^{-2}\text{A}$. The current will damage the razor since the current in Rome is nearly 2 times that in USA.

To determine

(c)

The type of transformer that is needed in Rome so as to operate the razor.

Answer to Problem 77QAP

A step down transformer with the number of turns in the primary equal to 2 times that in secondary needs to be used in Rome so as to operate the razor.

Explanation of Solution

Given:

  $V_{\text{rms,Rome}}=240\text{ V}$

  $V_{\text{rms,USA}}=120\text{ V}$

Formula used:

The applied voltage of 240 V must be brought down to 120 V for the razor to operate safely. The secondary voltage is equal to the voltage in USA and the primary is equal to the voltage in Rome.

  $\begin{array}{c}{V}_S=V_{\text{rms,USA}}\\ V_P=V_{\text{rms,Rome}}\end{array}$

Therefore,

  $\frac{V_{\text{rms,USA}}}{V_{\text{rms,Rome}}}=\frac{V_S}{V_P}=\frac{N_S}{N_P}$......(4)

Calculation:

Substitute the values of variables in equation (4) and calculate the transformer ratio.

  $\frac{N_S}{N_P}=\frac{120\text{ V}}{240\text{ V}}=\frac{1}{2}$

Since the number of turns in the primary is more than the number of turns in the secondary, the transformer is a step down transformer.

Conclusion:

Thus, a step down transformer with the number of turns in the primary equal to 2 times that in secondary needs to be used in Rome so as to operate the razor.

To determine

(d)

The resistance of the razor.

Answer to Problem 77QAP

The resistance of the razor is found to be 2880 Ω.

Explanation of Solution

Given:

  $P_{\text{USA}}=5.00\text{ W}$

  $V_{\text{rms,USA}}=120\text{ V}$

Formula used:

The resistance of the razor is given by,

  $R=\frac{V_{\text{rms,USA}}^2}{P_{\text{USA}}}$

Calculation:

Substitute the known values of the variables in the formula and calculate the resistance.

  $\begin{array}{c}R=\frac{V_{\text{rms,USA}}^2}{P_{\text{USA}}}\\ =\frac{{\left(120\text{ V}\right)}^2}{5.00\text{ W}}\\ =2880\Omega \end{array}$

Conclusion:

Thus, the resistance of the razor is found to be 2880 Ω.