In the circuit shown in Fig. E26.34, the 6.0 Ω resistor is consuming energy at a rate of 24 J/s when the current through it flows as shown. What are the polarity and emf ε of the unknown battery, assuming it has negligible internal resistance?

Verified step by step guidance

1. First, understand that the power consumed by the resistor is given by the formula: $P=I^{2}R$, where $P$ is the power, $I$ is the current, and $R$ is the resistance.
2. Given that the power $P$ is 24 J/s and the resistance $R$ is 6.0 Ω, use the formula to solve for the current $I$: $I=\sqrt{\frac{P}{R}}$.
3. Once you have the current, apply Kirchhoff's loop rule to the circuit. This rule states that the sum of the potential differences (voltage) around any closed loop in a circuit must be zero.
4. Identify the direction of the current flow and the polarity of the known battery in the circuit. Use the loop rule to write an equation that includes the emf $\epsilon$ of the unknown battery, the voltage drop across the resistor, and any other known voltages.
5. Solve the equation for the unknown emf $\epsilon$. The sign of $\epsilon$ will indicate the polarity of the unknown battery. If $\epsilon$ is positive, the polarity is in the direction of the assumed current flow; if negative, it is opposite.

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 essential for calculating the current and voltage in the circuit.

Power in Electrical Circuits

Power in electrical circuits is the rate at which energy is consumed or converted. It is calculated using the formula P = IV, where P is power, I is current, and V is voltage. Alternatively, it can be expressed as P = I^2R or P = V^2/R, depending on known values. Understanding power consumption helps determine the current flowing through the resistor.

Electromotive Force (EMF)

Electromotive force (EMF) is the energy provided by a battery or power source per coulomb of charge. It is the potential difference across the terminals of a battery when no current is flowing. EMF is crucial for determining the voltage supplied by the unknown battery in the circuit, influencing the current and power distribution.