Question

Using the exact exponential treatment, find how much time is required to charge an initially uncharged 100-pF capacitor through a 75.0-ΜΩ resistor to 90.0% of its final voltage.

a) 1.73x10
b) 7.50x10
c) 7.50×10-
d) 3.26×10-3

Answer 1

The time required to charge an initially uncharged 100-pF capacitor through a 75.0-MΩ resistor to 90% of its final voltage is 1.73 x 10-3 seconds, which is calculated using the charging formula for a capacitor and exponential functions.

Step-by-step explanation:

To calculate the time required to charge a capacitor through a resistor to a certain percentage of its final voltage, we use the formula for the charging of a capacitor, which involves an exponential function. The formula for the voltage across the capacitor at any time t during charging is V(t) = Vfinal(1 - e-t/RC), where Vfinal is the final voltage, R is the resistance, C is the capacitance, and e is the base of the natural logarithm.

To find the time when the voltage reaches 90.0% of its final value, we set V(t) = 0.90Vfinal and solve for t. In this case, with a 100-pF (100 x 10-12 F) capacitor and a 75.0-MΩ (75.0 x 106 Ω) resistor, we find that t = -RC ln(1 - 0.90). Plugging in the values, we have t = -(75.0 x 106 Ω)(100 x 10-12 F) ln(1 - 0.90) which simplifies to t = 1.73 x 10-3 seconds.