Question: A controller on an electronic arcade game consists of a variable resistor connected across the plates of a$0.220\mu F$capacitor. The capacitor is charged to 5.00 V, then discharged through the resistor. The time for the potential difference across the plates to decrease to 0.800 Vis measured by a clock inside the game. If the range of discharge times that can be handled effectively is from$10.0\mu s$to 6.00 ms, what should be the (a) lower value and (b) higher value of the resistance range of the resistor?

• a)The capacitance of the capacitor,$C=0.220\mu F=0,220\times {10}^{-6}F.$.
• b)The potential difference of capacitor,$V_0=5.00V$.
• c)The decrease in the potential difference of the plates,$V=0.800V.$ .
• d)Discharging time value range,$t=10\times {10}^{-6}sto6.0\times {10}^{-2}s$.

The simple concept of charging and discharging a given capacitor also gives us the idea that the potential drop for the same capacitor is directly dependent on the charge separation value at a given time. Again, a higher resistance implies a longer discharging time and vice-versa.

Formula:

The potential difference across a RC connection, $V=V_0\left(e^{-t/RC}\right)$ (i)

Now, using the given data in equation (i), the lower value of the resistance R range of the resistor can be given for the shorter discharging time as follows:

$R=\frac{t}{CIn(V_0/V)}$

Substitute the values in the above expression, and we get,

$$\begin{gathered} R=\frac{10\times {10}^{-6}s}{\left(0.0220\times {10}^{-6}F\right)In\left(5.0V/0.800V\right)} \\ =24.8\Omega \end{gathered}$$

Hence, the value of lower resistance is $24.8\Omega$.

Similarly, using the given data in equation (i), the higher value of the resistance range of the resistor can be given for the shorter discharging time as follows:

$R=\frac{t}{CIn\left(V_0/V\right)}$

Substitute the values in the above expression, and we get,

$$\begin{gathered} R=\frac{6\times {10}^{-2}s}{\left(0.0220\times {10}^{-6}F\right)In\left(5.0V/0.800V\right)} \\ =1.49\times {10}^4\Omega \end{gathered}$$

Hence, the value of lower resistance is$1.49\times {10}^4\Omega$ .