A 220 uF capacitor with a 10% tolerance rating is charged. The voltage across the capacitor is measured to be 110 +/- 5.5 volt. Calculate the electrical charge stored in the capacitor. Estimate the error in the charge by propagating the uncertainty in the capacitance and the voltage.
To calculate the electrical charge stored in a capacitor, you can use the formula: Q = C * V, where Q is the charge, C is the capacitance, and V is the voltage.
Given:
- Capacitance (C) = 220 uF (microfarads)
- Tolerance rating = 10%
- Voltage (V) = 110 +/- 5.5 V (volts)
Step 1: Calculate the capacitance range
The tolerance rating of 10% means that the actual capacitance of the capacitor can vary by plus or minus 10%.
To calculate the capacitance range, we can use the formula:
Capacitance range = Capacitance * tolerance/100
In this case:
Capacitance range = 220 uF * 10/100 = 22 uF
So, the capacitor's capacitance can vary between 198 uF and 242 uF (220 uF +/- 22 uF).
Step 2: Calculate the charge range
To calculate the charge range, we need to consider both the capacitance range and the voltage range.
The charge range can be calculated as follows:
Minimum charge = Minimum capacitance * Minimum voltage
Maximum charge = Maximum capacitance * Maximum voltage
In this case:
Minimum capacitance = 198 uF
Maximum capacitance = 242 uF
Minimum voltage = 110 V - 5.5 V = 104.5 V
Maximum voltage = 110 V + 5.5 V = 115.5 V
Minimum charge = 198 uF * 104.5 V = 20,691 uC (microcoulombs)
Maximum charge = 242 uF * 115.5 V = 27,996 uC (microcoulombs)
So, the charge stored in the capacitor can range from 20,691 uC to 27,996 uC.
Step 3: Estimate the error in the charge
To estimate the error in the charge, we need to propagate the uncertainty in both the capacitance and the voltage.
The error in the charge can be calculated using the formula:
Error in charge = (Max charge - Min charge) / 2
In this case:
Error in charge = (27,996 uC - 20,691 uC) / 2 = 3,652.5 uC (microcoulombs)
Therefore, the estimated error in the charge stored in the capacitor is approximately +/- 3,652.5 uC.