Chapter 21, Problem 101A

(a)

To determine

representation of the slop from given graph.

Answer to Problem 101A

The slope of the line represents capacitance.

Explanation of Solution

Given:

The given graph is shown below:

  

Formula used:

The slope of a line is expressed as

  $\text{slope}=\frac{\text{rise}}{\text{run}}$  ......(1)

Calculation:

The slope of given line from the graph is $\frac{q}{\Delta V}$ .

It is known that, capacitance $C$ is defined as $C=\frac{q}{\Delta V}$ .

Conclusion:

Hence, the slope of the line represents capacitance.

(b)

To determine

capacitance of the capacitor.

Answer to Problem 101A

The capacitance of the capacitor is $\boxed{C=0.5\text{μF}}$ .

Explanation of Solution

Given:

The given graph is shown below:

  

Formula used:

The slope of a line is expressed as

  $\text{slope}=\frac{\text{rise}}{\text{run}}$  ......(1)

Capacitance $C$ is defined as

  $C=\frac{q}{\Delta V}$  ......(2)

Calculation:

Below graph shows a plot of potential difference across x-axis and charge across y-axis.

  

In the above graph, as the potential difference increases, the changes increases linearly and thus the graph obtained will be straight line passing through the origin.

The slope of the graph gives the capacitance of the capacitor. That is,

  $m=\frac{y_1-y_2}{x_1-x_2}$  ......(3)

From the above graph, $x_1=20\text{V}$ , $x_2=10\text{V}$ , $y_1=10\text{μC}$ and $y_2=5\text{μC}$ .

So, the capacitance of the capacitor is,

  $\begin{array}{l}C=\frac{\left(10-5\right)\text{μC}}{\left(20-10\right)\text{V}}\\ \boxed{C=0.5\text{μF}}\end{array}$

Conclusion:

Hence, the capacitance of the capacitor is $\boxed{C=0.5\text{μF}}$ .

(c)

To determine

the area under the graph line.

Answer to Problem 101A

The area under the graph line represents the energy stored in a capacitor or the work done.

Explanation of Solution

Given:

The given graph is shown below:

  

Formula used:

The area under the graph line is given by

  $A=\frac{1}{2}\Delta V.q$  ......(1)

Capacitance $C$ is defined as

  $C=\frac{q}{\Delta V}$  ......(2)

Calculation:

Rearrange the equation (2),

  $q=C\Delta V$  ......(3)

Substitute the value of $q$ in equation (1),

  $A=\frac{1}{2}C\Delta V^2$  ......(4)

It is known that, the energy stored in a capacitor or the work done is defined as $W=\frac{1}{2}C\Delta V^2$  ......(5)

Therefore, the area under the graph line represents the energy stored in a capacitor or the work done.

Conclusion:

Hence, the area under the graph line represents the energy stored in a capacitor or the work done.