The duration of a photographic flash is related to an 𝑅𝐶 time constant, which is 0.100 microseconds for a certain camera. The capacitor in the camera is in a circuit with a charging resistance of 800 kiloohms, while the flash lamp of the camera has a resistance of 40.0 milliohms during discharge. What is the capacitance of the capacitor? What is the time constant for charging the capacitor?
This statement tells us this 𝑅𝐶 circuit’s time constant 0.100 microseconds, which we’ll call 𝜏. It also tells us that when the circuit is charging, it has a resistance of 800 kiloohms — a resistance value we will name 𝑅 sub 𝑐. And that while the circuit is discharging, its resistance is 40.0 milliohms — what we’ll call 𝑅 sub 𝑑.
In part one, we wanna solve for the capacitance of the capacitor. We’ll call that capital 𝐶. And in part two, we wanna solve for the time constant not for discharging, but for charging a capacitor — what we’ll 𝜏 sub 𝑐.
We can start off solving this problem by recalling the definition for a time constant for an 𝑅𝐶 circuit. In a 𝑅𝐶 circuit, the time constant, which we can call 𝜏, is equal to the product of the circuit’s resistance times its capacitance. In our case, we can write that 𝜏 is equal to 𝑅 sub 𝑑, the discharge resistance in our circuit, multiplied by 𝐶, the capacitance, or 𝐶 equals 𝜏 over 𝑅 sub 𝑑.
We’re told 𝜏 and 𝑅 sub 𝑑 in our problem statement. So we can plug in and solve for 𝐶. When we do, we’re careful to use units of seconds for our time constant and units of ohms for our resistance. When we calculate this fraction, we find that 𝐶 is 2.50 times 10 to the negative six farads or 2.50 microfarads. That’s the capacitance in our circuit whether charging or discharging.
Next, we wanna solve for 𝜏 sub 𝑐, which is the time constant of our circuit when it’s charging rather than discharging. We’ll again use our time constant relationship. But this time instead of using 𝑅 sub 𝑑, we’ll use 𝑅 sub 𝑐 — the effective resistance of our circuit when it charges.
We’re given 𝑅 sub 𝑐 in the statement and we’ve solved for 𝐶, the capacitance of our circuit, earlier. So we’re ready to plug in and solve for 𝜏 sub 𝑐. When we do, we’re sure to use units of ohms for resistance and farads for capacitance. Calculating this product, we find that 𝜏 sub 𝑐 is equal to 2.00 seconds. That’s the time constant of the circuit while it charges.