Math Problem Statement

Calculate the terminal voltage for a battery with an internal resistance of 0.685 Ω and an emf of 4.25 V when the battery is connected in series with an 77.6 Ω resistor.

Solution

To calculate the terminal voltage (VterminalV_{\text{terminal}}) of the battery, we use the following formula:

Vterminal=emfIrV_{\text{terminal}} = \text{emf} - I \cdot r

where:

  • emf\text{emf} is the electromotive force of the battery (4.25V4.25 \, \text{V}),
  • II is the current in the circuit,
  • rr is the internal resistance of the battery (0.685Ω0.685 \, \Omega).

Step 1: Calculate the total resistance of the circuit

The total resistance RtotalR_{\text{total}} is the sum of the internal resistance of the battery rr and the resistance of the external resistor RR:

Rtotal=r+RR_{\text{total}} = r + R

Rtotal=0.685Ω+77.6Ω=78.285ΩR_{\text{total}} = 0.685 \, \Omega + 77.6 \, \Omega = 78.285 \, \Omega

Step 2: Calculate the current in the circuit

Using Ohm's law (I=emfRtotalI = \frac{\text{emf}}{R_{\text{total}}}), the current is:

I=emfRtotalI = \frac{\text{emf}}{R_{\text{total}}}

I=4.25V78.285Ω0.0543A(54.3mA)I = \frac{4.25 \, \text{V}}{78.285 \, \Omega} \approx 0.0543 \, \text{A} \, (54.3 \, \text{mA})

Step 3: Calculate the terminal voltage

The terminal voltage is then calculated as:

Vterminal=emfIrV_{\text{terminal}} = \text{emf} - I \cdot r

Substituting the values:

Vterminal=4.25V(0.0543A0.685Ω)V_{\text{terminal}} = 4.25 \, \text{V} - (0.0543 \, \text{A} \cdot 0.685 \, \Omega)

Vterminal=4.25V0.0372VV_{\text{terminal}} = 4.25 \, \text{V} - 0.0372 \, \text{V}

Vterminal4.21VV_{\text{terminal}} \approx 4.21 \, \text{V}

Final Answer:

The terminal voltage of the battery is approximately 4.21 V.

Would you like further clarifications or have more related questions? Here are some related topics you might explore:

  1. How does increasing the internal resistance affect the terminal voltage?
  2. What happens to the current if the external resistance is reduced?
  3. Can you calculate power dissipated in the external resistor?
  4. How is terminal voltage related to energy loss within the battery?
  5. What would happen if the external resistor is replaced with a short circuit?

Tip: Always check the units for resistance, current, and voltage to ensure consistency in calculations.

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Math Problem Analysis

Mathematical Concepts

Ohm's Law
Series Circuits
Voltage Drop

Formulas

V_terminal = emf - I * r
I = emf / R_total
R_total = r + R

Theorems

Ohm's Law

Suitable Grade Level

Grades 9-12