Let's analyze the problem in the image step by step.

Problem Summary:

We are given:

1. The equation for the charging current: $i = i_0 e^{-t/(RC)}$
2. Parameters:
   • Resistance values ($R$): $1 \, \Omega$, $10 \, \Omega$, and $100 \, \Omega$
   • Capacitance ($C$): $5 \, \mu\text{F}$
   • Step potential: $100 \, \text{mV}$

Questions to solve:

1. Calculate the time constant ($RC$) for each resistance.
2. Calculate the initial charging current ($i_0$) for each cell.

Step 1: Time Constant Calculation ($RC$)

The time constant is given by: $RC = R \cdot C$ Where:

• $R$ is the resistance
• $C = 5 \, \mu\text{F} = 5 \times 10^{-6} \, \text{F}$

For each resistance:

1. $R = 1 \, \Omega$: $RC = 1 \cdot 5 \times 10^{-6} = 5 \times 10^{-6} \, \text{s}$
2. $R = 10 \, \Omega$: $RC = 10 \cdot 5 \times 10^{-6} = 5 \times 10^{-5} \, \text{s}$
3. $R = 100 \, \Omega$: $RC = 100 \cdot 5 \times 10^{-6} = 5 \times 10^{-4} \, \text{s}$

Thus, the time constants are:

• $5 \, \mu\text{s}$ for $1 \, \Omega$
• $50 \, \mu\text{s}$ for $10 \, \Omega$
• $500 \, \mu\text{s}$ for $100 \, \Omega$

Step 2: Initial Current ($i_0$)

The initial charging current is given by: $i_0 = \frac{\Delta V}{R}$ Where:

• $\Delta V = 100 \, \text{mV} = 0.1 \, \text{V}$

For each resistance:

1. $R = 1 \, \Omega$: $i_0 = \frac{0.1}{1} = 0.1 \, \text{A} \, (100 \, \text{mA})$
2. $R = 10 \, \Omega$: $i_0 = \frac{0.1}{10} = 0.01 \, \text{A} \, (10 \, \text{mA})$
3. $R = 100 \, \Omega$: $i_0 = \frac{0.1}{100} = 0.001 \, \text{A} \, (1 \, \text{mA})$

Thus, the initial charging currents are:

• $100 \, \text{mA}$ for $1 \, \Omega$
• $10 \, \text{mA}$ for $10 \, \Omega$
• $1 \, \text{mA}$ for $100 \, \Omega$

Final Results:

1. Time constants ($RC$):

   • $1 \, \Omega$: $5 \, \mu\text{s}$
   • $10 \, \Omega$: $50 \, \mu\text{s}$
   • $100 \, \Omega$: $500 \, \mu\text{s}$

2. Initial currents ($i_0$):

   • $1 \, \Omega$: $100 \, \text{mA}$
   • $10 \, \Omega$: $10 \, \text{mA}$
   • $100 \, \Omega$: $1 \, \text{mA}$

Let me know if you'd like a deeper explanation of any step or more related calculations! 😊

Related Questions:

1. How does the time constant affect the charging and discharging process of a capacitor?
2. What is the significance of the exponential decay in the current equation?
3. How would increasing the capacitance affect the results?
4. Why is the initial current inversely proportional to the resistance?
5. Can you derive the time-dependent current equation from basic capacitor principles?

Tip:

Always check the units carefully when calculating values involving microfarads ($\mu\text{F}$) and millivolts ($\text{mV}$) to avoid scaling errors!