Q67P - Fundamentals Of Physics

Question

A 500 W heating unit is designed to operate with an applied potential difference of 115 V . (a) By what percentage will its heat output drop if the applied potential difference drops to 110 V ? Assume no change in resistance. (b) If you took the variation of resistance with temperature into account, would the actual drop in heat output be larger or smaller than that calculated in (a)?

Step-by-Step Solution

Answer

1. The percentage that the unit will heat output drops if the applied potential difference drops to 110 V be  -8.6% .
2. If variation in resistance with temperature is taken into consideration than the actual drop in   will be smaller. 

Step 1: The given data:

The power of the heating unit,  P = 500 W

Initial potential difference, ${\mathrm{V}}_1=115 \mathrm{V}$

The new potential difference, ${\mathrm{V}}_2=110 \mathrm{V}$

Step 2: Understanding the concept of the power and potential difference:

The force is proportional to the square of the potential. From this calculate the percentage change in performance. As temperature decreases, resistance also decreases, but power is inversely proportional to resistance, so as resistance decreases, power increases. Thus, the actual drop in   will be smaller if you take into account the temperature dependence of the resistance.

Formula:

The power generated due to a potential difference in a battery, 

$\mathrm{P}=\frac{{\mathrm{V}}^2}{\mathrm{R}}$                                                                                                                  ….. (1)

Here, V is the potential difference and R is the resistance.

Step 3: (a) Calculation of the percentage of the heating element to have a potential drop of 110 V :

From equation (1), you can get that power is directly proportional to the square of the potential difference. Thus, it gives 

 

 $∆\mathrm{P}\infty ∆{\mathrm{V}}^2$

Now, the differential equation of the power using the above condition can be given as:

$∆\mathrm{P}=2\mathrm{V}∆\mathrm{V}$ 

Where, $∆\mathrm{P}$ is the change in power and $∆\mathrm{V}$ is the change in the potential.

 

Thus, you can calculate percentage change in power as follows:

$$\begin{gathered} ∆\mathrm{P}/\mathrm{P}=2∆\mathrm{V}/\mathrm{V} \\ =\frac{2\left({\mathrm{V}}_2-{\mathrm{V}}_1\right)}{{\mathrm{V}}_1} \end{gathered}$$ 

Substitute known valued in the above equation.

 $$\begin{gathered} ∆\mathrm{P}/\mathrm{P}=\frac{2\left(110-115\right)}{110} \\ =-0.086 \\ =-8.6\% \end{gathered}$$

Hence, the value of the percentage is 8.6 %.

Step 4: (b) Calculation of the behavior of the actual drop in the output heat:

At lower temperature, R also decreases, but P is inversely proportional to resistance considering equation (1).

Thus, as resistance decreases, power increases. i.e.

$\mathrm{P}\infty {\mathrm{R}}^{-1}$  

 

Hence, the actual drop in P will be smaller when the temperature dependence of the resistance taken in to consideration.