a) What is the maximum current in a 4.00 µF capacitor when it is connected across a North American electrical outlet having ΔVrms = 120 V, f = 60.0 Hz?
mA
(b) What is the maximum current when it is connected across a European electrical outlet having ΔVrms = 240 V and f = 50.0 Hz?
a. Xc = 1/2pi*F*c
Xc = 1/(6.28*60*4*10^-6) = 663.5 Ohms.
I = V/Xc = 120/663.5 = 0.181A rms.
b. Same procedure as "a".
To find the maximum current in a capacitor when connected across an electrical outlet, we can use the formula:
Imax = (ΔVrms * 2π * f) / Xc
Where:
Imax is the maximum current
ΔVrms is the root mean square voltage
f is the frequency
Xc is the capacitive reactance
To solve for (a) using a North American electrical outlet:
a) Given:
ΔVrms = 120 V
f = 60.0 Hz
C = 4.00 µF
First, we need to find the capacitive reactance (Xc):
Xc = 1 / (2π * f * C)
Substituting the given values:
Xc = 1 / (2π * 60.0 Hz * 4.00 µF)
Xc = 1 / (377 rad/s * 4.00 × 10^(-6) F)
Xc ≈ 663.42 Ω
Now, we can substitute the values back into the formula to find the maximum current (Imax):
Imax = (ΔVrms * 2π * f) / Xc
Imax = (120 V * 2π * 60.0 Hz) / 663.42 Ω
Imax ≈ 0.545 A
Therefore, the maximum current in a 4.00 µF capacitor when connected across a North American electrical outlet is approximately 0.545 A (mA).
Now, let's solve (b) using a European electrical outlet:
b) Given:
ΔVrms = 240 V
f = 50.0 Hz
C = 4.00 µF
Using the same process as in (a), we find the capacitive reactance (Xc):
Xc = 1 / (2π * f * C)
Xc = 1 / (2π * 50.0 Hz * 4.00 µF)
Xc = 1 / (314 rad/s * 4.00 × 10^(-6) F)
Xc ≈ 795.77 Ω
Now, substitute the values into the formula to find the maximum current (Imax):
Imax = (ΔVrms * 2π * f) / Xc
Imax = (240 V * 2π * 50.0 Hz) / 795.77 Ω
Imax ≈ 1.904 A
Therefore, the maximum current in a 4.00 µF capacitor when connected across a European electrical outlet is approximately 1.904 A.
To find the maximum current in a capacitor, we can use the formula:
Imax = ΔVrms * 2πf * C
Where:
- ΔVrms is the root mean square voltage across the capacitor
- f is the frequency in hertz (Hz)
- C is the capacitance in farads (F)
Now, let's calculate the maximum current in both scenarios:
a) For the North American electrical outlet:
- ΔVrms = 120 V
- f = 60.0 Hz
- C = 4.00 µF = 4.00 * 10^(-6) F
Using the formula, we have:
Imax = 120 V * 2π * 60.0 Hz * 4.00 * 10^(-6) F
To simplify the calculation, let's convert the units:
Imax = 120 V * 2π * 60.0 s^(-1) * 4.00 * 10^(-6) F
Imax = 0.018 A
Therefore, the maximum current in a 4.00 µF capacitor connected across a North American electrical outlet is 0.018 A (or 18 mA).
b) For the European electrical outlet:
- ΔVrms = 240 V
- f = 50.0 Hz
- C = 4.00 µF = 4.00 * 10^(-6) F
Using the formula, we have:
Imax = 240 V * 2π * 50.0 Hz * 4.00 * 10^(-6) F
To simplify the calculation, let's convert the units:
Imax = 240 V * 2π * 50.0 s^(-1) * 4.00 * 10^(-6) F
Imax = 0.048 A
Therefore, the maximum current in a 4.00 µF capacitor connected across a European electrical outlet is 0.048 A (or 48 mA).