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I am trying to figure out where I went wrong on the following problem:

schematic

simulate this circuit – Schematic created using CircuitLab

The two batteries are identical, and each has an open-circuit voltage of 1.5V. The lamp has a resistance of 5\$\Omega\$ when lit. With the switch closed, 2.5V is measured across the lamp. What is the internal resistance of each battery?

(Problem 2.1 in Agarwal and Lang's, Foundations of Analog and Digital Electronic Circuits). Note the answer printed in the back of the book: 0.5\$\Omega\$ .

Here's my solution:

Step 1

Use element law to find current, \$ {i}_{1} \$, through bulb. $$ v=iR \rightarrow {i}_{1} = \frac{v}{{R}_{bulb}}=\frac{2.5V}{5\Omega}=\frac{1}{2}A. $$

Step 2

Model the internal resistance of each battery as a resistor. State the equivalent resistance of the two resistors in series. $$ {R}_{eq}={R}_{1}+{R}_{2}=2{R}_{n} $$

Step 3

By Kirchoff's Voltage Law, the potential difference across the two batteries must be equal and opposite to the potential difference across the lamp. I combine the element law with the above expression in the following manner: $$ v={i}_{2}{R}_{eq} \rightarrow {R}_{n}=\frac{1}{2}\frac{v}{{i}_{2}} (eqn. 1) $$

Step 4

By Kirchoff's Current Law, the sum of currents at any node is zero. $$ {i}_{1}-{i}_{2}=0 \rightarrow {i}_{2}={i}_{1} (eqn.2) $$

Step 5

Combine eqns. 1 & 2 to find \${R}_{n}\$, the internal resistance of a single battery. $$ {R}_{n}=\frac{1}{2}\frac{v}{{i}_{1}}=2.5\Omega $$

Conclusion

After reflecting on the problem statement, especially the open-circuit voltage part, I know that I am committing some logical fallacy. However I just cannot see it on my own. Where did I go wrong? Should I not imagine that the internal resistance of the batteries can be modeled as a resistor? Would an energy / power approach be better suited to this problem?

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    \$\begingroup\$ technically, a battery is a group of connected cells. you have 2 cells forming a 3V battery. you already figured out the current. ... the internal resistance of the battery is using up 0.5V at 0.5A, which makes it 1Ω ... evenly divided between 2 cells .... you figure out the rest \$\endgroup\$ – jsotola Jan 9 '18 at 6:30
  • \$\begingroup\$ OP got lost after step 2. \$\endgroup\$ – Sparky256 Jan 9 '18 at 6:35
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    \$\begingroup\$ `It is nice to see someone asking a homework question where they clearly show they have made an effort, and attempted to understand where they went wrong. Very refreshing. +1 \$\endgroup\$ – MCG Jan 9 '18 at 8:30
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I think your misconception happens in step 3:

By Kirchoff's Voltage Law, the potential difference across the two batteries must be equal and opposite to the potential difference across the lamp. I combine the element law with the above expression in the following manner [...]

This is not true or at least not written precisely enough. Maybe you should draw the complete circuit to make it easier to understand:

schematic

simulate this circuit – Schematic created using CircuitLab

Now apply the voltage law:

$$V(BAT1) + (-I\times R1) + V(BAT2) + (-I\times R2) + (-I\times R(LAMP1)) = 0$$ $$2 V_{bat} - I\times 5\ \Omega = 2 I\times X$$ $$\frac{V_{bat}}{I} - \frac{1}{2}\times 5\ \Omega = X$$ $$\frac{1,5\ V}{0,5\ A} - \frac{1}{2}\times 5\ \Omega = X$$ $$X = 0.5\ \Omega$$

I omitted the current through the voltage meter (assumed to be ideal), so no need to apply current law as only one known current is flowing in the loop.

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You made it too complicated. The battery current is 0.5A, as you said. That 0.5A is causing a 0.5V drop in the battery voltage due to the combined series resistance of the batteries. We can just use Ohm's law. Vdrop = Ibatt * Rbatt.

So, Rbatt = 0.5V / 0.5A = 1 Ohm. But that is the combined series resistance. So each battery contributes 0.5 Ohms to the total.

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  • \$\begingroup\$ short and sweet. (non-electrical short) \$\endgroup\$ – richard1941 Jan 11 '18 at 18:36
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The mistake in the analysis is in equation 1. The correct equation is,

\$V_{Bat1}+V_{Bat2}-i_2R_{eq}=v\$

On a side note, The internal resistance arises due to mobility of electrolyte, the concentration , the surface area of the electrodes and the length between the electrodes. The voltage arises due to redox potentials of the electrodes and there is the nernst equation for the concentration.

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