Resistor–Capacitor (RC) Circuits - AP Physics C: Electricity and Magnetism
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What is the initial charge on a capacitor in an RC circuit?
What is the initial charge on a capacitor in an RC circuit?
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Zero when starting to charge. Uncharged capacitor acts like open circuit initially.
Zero when starting to charge. Uncharged capacitor acts like open circuit initially.
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Identify the unit of capacitance.
Identify the unit of capacitance.
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Farad (F). Named after Michael Faraday, measures charge storage ability.
Farad (F). Named after Michael Faraday, measures charge storage ability.
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State the formula for current in a discharging RC circuit.
State the formula for current in a discharging RC circuit.
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$I(t) = I_0 e^{-t/\tau}$. Current decays exponentially from initial value $I_0$ during discharge.
$I(t) = I_0 e^{-t/\tau}$. Current decays exponentially from initial value $I_0$ during discharge.
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Calculate voltage across a capacitor at $t = 4\tau$ when discharging.
Calculate voltage across a capacitor at $t = 4\tau$ when discharging.
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$V(t) \approx 0.018 V_0$. At $t=4\tau$: $e^{-4} \approx 0.018$ of initial voltage remains.
$V(t) \approx 0.018 V_0$. At $t=4\tau$: $e^{-4} \approx 0.018$ of initial voltage remains.
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State the formula for the voltage across a charging capacitor in an RC circuit.
State the formula for the voltage across a charging capacitor in an RC circuit.
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$V(t) = V_0(1 - e^{-t/\tau})$. Exponential approach to final voltage $V_0$ with time constant $\tau$.
$V(t) = V_0(1 - e^{-t/\tau})$. Exponential approach to final voltage $V_0$ with time constant $\tau$.
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What is the final voltage across a fully charged capacitor in an RC circuit?
What is the final voltage across a fully charged capacitor in an RC circuit?
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Equal to the supply voltage $V_0$. Capacitor voltage approaches source voltage as charging completes.
Equal to the supply voltage $V_0$. Capacitor voltage approaches source voltage as charging completes.
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Find the time required to reach half the maximum charge in an RC circuit.
Find the time required to reach half the maximum charge in an RC circuit.
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$t = \tau \ln(2)$. Natural logarithm of 2 times the time constant gives half-charge time.
$t = \tau \ln(2)$. Natural logarithm of 2 times the time constant gives half-charge time.
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Determine $I(0)$ for an RC circuit with $R = 10 \Omega$ and $V_0 = 10 \text{V}$.
Determine $I(0)$ for an RC circuit with $R = 10 \Omega$ and $V_0 = 10 \text{V}$.
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$I(0) = 1\text{A}$. Using Ohm's law: $I = V/R = 10/10 = 1$ ampere.
$I(0) = 1\text{A}$. Using Ohm's law: $I = V/R = 10/10 = 1$ ampere.
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Calculate $I(t)$ at $t = \tau$ for a discharging circuit if $I_0 = 2 \text{A}$.
Calculate $I(t)$ at $t = \tau$ for a discharging circuit if $I_0 = 2 \text{A}$.
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$I(t) = 0.736 \text{A}$. At $t=\tau$: $I = I_0 e^{-1} = 2 \times 0.368 = 0.736$ A.
$I(t) = 0.736 \text{A}$. At $t=\tau$: $I = I_0 e^{-1} = 2 \times 0.368 = 0.736$ A.
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What is the relationship between $V(t)$ and $V_0$ at $t = 2\tau$ when charging?
What is the relationship between $V(t)$ and $V_0$ at $t = 2\tau$ when charging?
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$V(t) = 0.865 V_0$. At $t=2\tau$: $1 - e^{-2} = 1 - 0.135 = 0.865$ of final voltage.
$V(t) = 0.865 V_0$. At $t=2\tau$: $1 - e^{-2} = 1 - 0.135 = 0.865$ of final voltage.
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State the formula for current in a discharging RC circuit.
State the formula for current in a discharging RC circuit.
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$I(t) = I_0 e^{-t/\tau}$. Current decays exponentially from initial value $I_0$ during discharge.
$I(t) = I_0 e^{-t/\tau}$. Current decays exponentially from initial value $I_0$ during discharge.
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What happens to the current in an RC circuit over time when charging?
What happens to the current in an RC circuit over time when charging?
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The current decreases exponentially. Current starts at maximum and decreases as capacitor voltage increases.
The current decreases exponentially. Current starts at maximum and decreases as capacitor voltage increases.
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What fraction of the initial voltage remains after $5\tau$ in an RC circuit?
What fraction of the initial voltage remains after $5\tau$ in an RC circuit?
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Less than 1%. After $5\tau$: $e^{-5} \approx 0.007$ or about 0.7% remains.
Less than 1%. After $5\tau$: $e^{-5} \approx 0.007$ or about 0.7% remains.
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Find $\tau$ for an RC circuit with $R = 5 \Omega$ and $C = 2 \text{F}$.
Find $\tau$ for an RC circuit with $R = 5 \Omega$ and $C = 2 \text{F}$.
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$\tau = 10 \text{s}$. Using $\tau = RC$: $5 \times 2 = 10$ seconds.
$\tau = 10 \text{s}$. Using $\tau = RC$: $5 \times 2 = 10$ seconds.
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What is the initial current in an RC circuit when the capacitor starts charging?
What is the initial current in an RC circuit when the capacitor starts charging?
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$I_0 = \frac{V_0}{R}$. Using Ohm's law: initial voltage across resistor equals supply voltage.
$I_0 = \frac{V_0}{R}$. Using Ohm's law: initial voltage across resistor equals supply voltage.
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Calculate the energy stored in a capacitor with $C = 4\text{F}$ and $V = 3\text{V}$.
Calculate the energy stored in a capacitor with $C = 4\text{F}$ and $V = 3\text{V}$.
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$U = 18\text{J}$. Using energy formula: $U = \frac{1}{2} \times 4 \times 3^2 = 18$ J.
$U = 18\text{J}$. Using energy formula: $U = \frac{1}{2} \times 4 \times 3^2 = 18$ J.
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Find $Q(t)$ at $t = \tau$ for a charging capacitor if $C = 1 \text{F}$, $V_0 = 5 \text{V}$.
Find $Q(t)$ at $t = \tau$ for a charging capacitor if $C = 1 \text{F}$, $V_0 = 5 \text{V}$.
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$Q(t) = 3.16 \text{C}$. At $t=\tau$: $Q = 1 \times 5 \times (1-e^{-1}) = 3.16$ C.
$Q(t) = 3.16 \text{C}$. At $t=\tau$: $Q = 1 \times 5 \times (1-e^{-1}) = 3.16$ C.
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Find the time at which $V(t) = 0.9V_0$ in a charging RC circuit.
Find the time at which $V(t) = 0.9V_0$ in a charging RC circuit.
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$t \approx 2.303\tau$. Solving $0.9 = 1-e^{-t/\tau}$ gives $t = -\tau \ln(0.1) = 2.303\tau$.
$t \approx 2.303\tau$. Solving $0.9 = 1-e^{-t/\tau}$ gives $t = -\tau \ln(0.1) = 2.303\tau$.
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What is the charge on a capacitor at any time $t$ during discharging?
What is the charge on a capacitor at any time $t$ during discharging?
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$Q(t) = Q_0 e^{-t/\tau}$. Charge decays exponentially from initial value $Q_0$ during discharge.
$Q(t) = Q_0 e^{-t/\tau}$. Charge decays exponentially from initial value $Q_0$ during discharge.
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Calculate the voltage across a capacitor at $t = \tau$ when charging.
Calculate the voltage across a capacitor at $t = \tau$ when charging.
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$V(t) = 0.632 V_0$. At one time constant: $1 - e^{-1} = 0.632$ of final voltage.
$V(t) = 0.632 V_0$. At one time constant: $1 - e^{-1} = 0.632$ of final voltage.
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What is the time constant $\tau$ for $R = 100 \Omega$ and $C = 10 \mu\text{F}$?
What is the time constant $\tau$ for $R = 100 \Omega$ and $C = 10 \mu\text{F}$?
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$\tau = 1 \text{ms}$. Converting units: $100 \times 10 \times 10^{-6} = 1 \times 10^{-3}$ s.
$\tau = 1 \text{ms}$. Converting units: $100 \times 10 \times 10^{-6} = 1 \times 10^{-3}$ s.
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Calculate the voltage across a capacitor at $t = 0$ when discharging.
Calculate the voltage across a capacitor at $t = 0$ when discharging.
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$V(t) = V_0$. At $t=0$ for discharging, capacitor starts at full voltage.
$V(t) = V_0$. At $t=0$ for discharging, capacitor starts at full voltage.
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What is the final charge on a capacitor in an RC circuit after full charge?
What is the final charge on a capacitor in an RC circuit after full charge?
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$Q = C \times V_0$. Final charge equals capacitance times applied voltage.
$Q = C \times V_0$. Final charge equals capacitance times applied voltage.
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What is the relationship between $V(t)$ and $V_0$ at $t = 3\tau$ when discharging?
What is the relationship between $V(t)$ and $V_0$ at $t = 3\tau$ when discharging?
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$V(t) = 0.05 V_0$. At $t=3\tau$: $e^{-3} = 0.05$ of initial voltage remains.
$V(t) = 0.05 V_0$. At $t=3\tau$: $e^{-3} = 0.05$ of initial voltage remains.
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What is the charge on a capacitor at any time $t$ during charging?
What is the charge on a capacitor at any time $t$ during charging?
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$Q(t) = C V_0 (1 - e^{-t/\tau})$. Charge accumulates exponentially approaching final value $CV_0$.
$Q(t) = C V_0 (1 - e^{-t/\tau})$. Charge accumulates exponentially approaching final value $CV_0$.
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Determine the time constant if $R = 250 \Omega$ and $C = 20 \mu\text{F}$.
Determine the time constant if $R = 250 \Omega$ and $C = 20 \mu\text{F}$.
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$\tau = 5 \text{ms}$. Converting units: $250 \times 20 \times 10^{-6} = 5 \times 10^{-3}$ s.
$\tau = 5 \text{ms}$. Converting units: $250 \times 20 \times 10^{-6} = 5 \times 10^{-3}$ s.
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What is the effect of doubling resistance on $\tau$ in an RC circuit?
What is the effect of doubling resistance on $\tau$ in an RC circuit?
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$\tau$ doubles. Time constant is directly proportional to resistance.
$\tau$ doubles. Time constant is directly proportional to resistance.
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What does the time constant $\tau$ represent in an RC circuit?
What does the time constant $\tau$ represent in an RC circuit?
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Time to reach approximately 63.2% of final value. At $t=\tau$, exponential function yields $1-e^{-1} \approx 0.632$ or 63.2%.
Time to reach approximately 63.2% of final value. At $t=\tau$, exponential function yields $1-e^{-1} \approx 0.632$ or 63.2%.
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State the formula for the voltage across a discharging capacitor in an RC circuit.
State the formula for the voltage across a discharging capacitor in an RC circuit.
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$V(t) = V_0 e^{-t/\tau}$. Exponential decay from initial voltage $V_0$ with time constant $\tau$.
$V(t) = V_0 e^{-t/\tau}$. Exponential decay from initial voltage $V_0$ with time constant $\tau$.
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What is the expression for $\tau$ in terms of resistance and capacitance?
What is the expression for $\tau$ in terms of resistance and capacitance?
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$\tau = R \times C$. Direct product of circuit resistance and capacitance values.
$\tau = R \times C$. Direct product of circuit resistance and capacitance values.
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