Final answer:
To calculate the time taken for a capacitor to reach 110 V, we apply the formula for the voltage across a charging capacitor and solve for time using the resistor and capacitor values given. The result is approximately 2721.3 seconds.
Step-by-step explanation:
The student is asking about the charging of a capacitor in an RC circuit, which is a physics concept related to electricity and circuits. To find out how long it takes the capacitor voltage to reach 110 V, we use the formula for the voltage across a charging capacitor:
V(t) = V0(1 - e-t/RC)
Where:
- V(t) is the voltage across the capacitor at time t
- V0 is the initial voltage across the capacitor (250 V in this case)
- R is the resistance (470 kΩ)
- C is the capacitance (10 μF)
- t is the time in seconds
- e is the base of the natural logarithm (approximately 2.71828)
To find the time t when the voltage is 110 V, we can rearrange the formula to solve for t:
t = -RC ln(1 - ⅓)
The RC time constant (τ) for the circuit is R multiplied by C:
τ = R ⋅ C
τ = 470 kΩ ⋅ 10 μF = 4700 seconds
Now, plug the values into the rearranged formula:
t = -4700 s ⋅ ln(1 - 110 V / 250 V)
t ≈ 4700 s ⋅ ln(1 -0.44)
t ≈ 4700 s ⋅ ln(0.56)
t ≈ 4700 s ⋅ (-0.579)
t ≈ 2721.3 s
Therefore, it takes approximately 2721.3 seconds for the capacitor voltage to reach 110 V.