When a resistor with resistance $R$ is connected to a $1.50$-V flashlight battery, the resistor consumes $0.0625$ W of electrical power. (Throughout, assume that each battery has negligible internal resistance.) What power does the resistor consume if it is connected to a $12.6$-V car battery? Assume that $R$ remains constant when the power consumption changes.

Verified step by step guidance

1. Start by recalling the formula for electrical power consumed by a resistor: P = V^2 / R, where P is the power, V is the voltage, and R is the resistance.
2. Given the initial conditions with the flashlight battery: P = 0.0625 W and V = 1.50 V, use the formula to solve for the resistance R. Rearrange the formula to R = V^2 / P.
3. Substitute the known values into the rearranged formula: R = (1.50^2) / 0.0625.
4. Now, use the resistance R calculated from the flashlight battery scenario to find the power consumed when connected to the car battery with a voltage of 12.6 V. Use the formula P = V^2 / R again.
5. Substitute the car battery voltage and the previously calculated resistance into the formula: P = (12.6^2) / R, where R is the resistance found in step 3.

Key Concepts

Ohm's Law

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points, and inversely proportional to the resistance. It is expressed as V = IR, where V is voltage, I is current, and R is resistance. This law is fundamental in calculating the current and understanding how voltage and resistance affect power consumption.

Power in Electrical Circuits

Electrical power is the rate at which electrical energy is transferred by an electric circuit. The power consumed by a resistor can be calculated using the formula P = VI, where P is power, V is voltage, and I is current. Alternatively, it can be expressed as P = V^2/R when the resistance is constant, which is useful for determining power consumption with different voltage sources.

Voltage Source Impact

The voltage source impacts the power consumed by a resistor. When the voltage increases, assuming resistance remains constant, the power consumed by the resistor increases quadratically according to the formula P = V^2/R. Understanding how different voltage sources affect power consumption is crucial for predicting changes in electrical circuits.