A typical small flashlight contains two batteries, each having an emf of 1.5 V, connected in series with a bulb having resistance 17 Ω. The resistance of real batteries increases as they run down. If the initial internal resistance is negligible, what is the combined internal resistance of both batteries when the power to the bulb has decreased to half its initial value?

Verified step by step guidance

1. Determine the initial power delivered to the bulb using the formula for power: $P=\frac{V^2}{R}$, where $V$ is the total emf of the batteries and $R$ is the resistance of the bulb.
2. Express the total emf of the batteries as $V=1.5+1.5=3.0$ V, and substitute this value along with the bulb resistance $R=17$ Ω into the power formula to calculate the initial power.
3. When the power decreases to half its initial value, the new power is $P=\frac{1}{2}P{\mathrm{initial}}_{}$. Use the modified power formula $P=\frac{V^2}{R}$, where $r$ is the combined internal resistance of the batteries, to relate the new power to the total resistance.
4. Set up the equation for the new power: $\frac{1}{2}P{\mathrm{initial}}_{}=\frac{V^2}{R}$. Substitute the known values for $V$, $R$, and $P{\mathrm{initial}}_{}$ into the equation.
5. Solve for the internal resistance $r$ by isolating it in the equation. Rearrange the terms to express $r$ in terms of the known quantities, and simplify the expression to find the combined internal resistance of the batteries.

Key Concepts

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed mathematically as V = IR. Understanding this law is crucial for analyzing circuits, as it helps determine how voltage, current, and resistance interact within the system.

Power in Electrical Circuits

The power (P) consumed by an electrical device is defined as the rate at which energy is used or converted. In a circuit, power can be calculated using the formula P = IV, where I is the current and V is the voltage. Additionally, power can also be expressed in terms of resistance as P = I²R or P = V²/R. This concept is essential for understanding how changes in resistance or current affect the power delivered to components like bulbs.

Internal Resistance of Batteries

Internal resistance refers to the resistance within a battery that opposes the flow of current. As batteries discharge, their internal resistance can increase, affecting the voltage available to external circuits. In the context of the flashlight, the internal resistance of the batteries becomes significant when the power to the bulb decreases, as it impacts the overall voltage and current supplied to the bulb, ultimately influencing its brightness.