c. For a charging RC circuit, after about how many time constants is the current down to 25% of its maximum?
c. For a charging RC circuit, after about how many time constants is the current down to 25% of its maximum?
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![**Question:**
For a charging RC circuit, after about how many time constants is the current down to 25% of its maximum?
**Explanation:**
In an RC (resistor-capacitor) circuit, the time constant, denoted by the symbol τ (tau), is defined as τ = RC, where R is resistance and C is capacitance. The time constant signifies the time required for the current or voltage to rise to approximately 63.2% of its maximum value or to drop to approximately 36.8% for a discharging capacitor.
In this context, we are asked when the current will decrease to 25% of its maximum value during the charging phase.
The formula for the charging current (I) in an RC circuit as a function of time (t) is given by:
\[ I(t) = I_0 (1 - e^{-t/τ}) \]
Where:
- \( I_0 \) is the initial current,
- \( e \) is the base of the natural logarithm,
- \( τ \) is the time constant.
Setting \( I(t) = 0.25 I_0 \):
\[ 0.25 I_0 = I_0 (1 - e^{-t/τ}) \]
Solving this equation will determine the time at which the current is 25% of the maximum.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2520f39-8b3c-4b2b-be15-831ac0e5241f%2F654cd52d-c500-45e1-9ec8-d73d0fd6e55d%2Fhyd1je_processed.png&w=3840&q=75)
Transcribed Image Text:**Question:**
For a charging RC circuit, after about how many time constants is the current down to 25% of its maximum?
**Explanation:**
In an RC (resistor-capacitor) circuit, the time constant, denoted by the symbol τ (tau), is defined as τ = RC, where R is resistance and C is capacitance. The time constant signifies the time required for the current or voltage to rise to approximately 63.2% of its maximum value or to drop to approximately 36.8% for a discharging capacitor.
In this context, we are asked when the current will decrease to 25% of its maximum value during the charging phase.
The formula for the charging current (I) in an RC circuit as a function of time (t) is given by:
\[ I(t) = I_0 (1 - e^{-t/τ}) \]
Where:
- \( I_0 \) is the initial current,
- \( e \) is the base of the natural logarithm,
- \( τ \) is the time constant.
Setting \( I(t) = 0.25 I_0 \):
\[ 0.25 I_0 = I_0 (1 - e^{-t/τ}) \]
Solving this equation will determine the time at which the current is 25% of the maximum.
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