48. Suppose the circuit shown in Example Problem 4 has these values: R_(1)=255Omega ,R_(2)=290Omega and Delta V_(1)=17V No other information is available. a. What is the current in the circuit? b. What is the potential difference across the battery? c. What is the total power used in the circuit, and what is the power used in each resistor? d. Does the sum of the power used in each resistor in the circuit equal the total power used in the circuit? Explain.

To solve this problem, we need to use Ohm's law and the power formula.<br /><br />Given information:<br />- $R_{1} = 255\Omega$<br />- $R_{2} = 290\Omega$<br />- $\Delta V_{1} = 17V$<br /><br />a. What is the current in the circuit?<br /><br />To find the current in the circuit, we can use Ohm's law:<br />$I = \frac{\Delta V_{1}}{R_{1} + R_{2}}$<br /><br />Substituting the given values:<br />$I = \frac{17V}{255\Omega + 290\Omega}$<br />$I = \frac{17V}{545\Omega}$<br />$I = 0.031A$<br /><br />Therefore, the current in the circuit is 0.031A.<br /><br />b. What is the potential difference across the battery?<br /><br />To find the potential difference across the battery, we can use the fact that the sum of the potential differences across each resistor is equal to the potential difference across the battery.<br /><br />$\Delta V_{\text{battery}} = \Delta V_{1} + \Delta V_{2}$<br /><br />Since the current is the same through both resistors, we can use Ohm's law to find the potential difference across the second resistor:<br />$\Delta V_{2} = I \cdot R_{2}$<br />$\Delta V_{2} = 0.031A \cdot 290\Omega$<br />$\Delta V_{2} = 8.99V$<br /><br />Therefore, the potential difference across the battery is:<br />$\Delta V_{\text{battery}} = 17V + 8.99V = 25.99V$<br /><br />c. What is the total power used in the circuit, and what is the power used in each resistor?<br /><br />To find the total power used in the circuit, we can use the power formula:<br />$P_{\text{total}} = \Delta V_{\text{battery}} \cdot I$<br />$P_{\text{total}} = 25.99V \cdot 0.031A$<br />$P_{\text{total}} = 0.811W$<br /><br />To find the power used in each resistor, we can use the power formula:<br />$P_{1} = I^{2} \cdot R_{1}$<br />$P_{1} = (0.031A)^{2} \cdot 255\Omega$<br />$P_{1} = 0.246W$<br /><br />$P_{2} = I^{2} \cdot R_{2}$<br />$P_{2} = (0.031A)^{2} \cdot 290\Omega$<br />$P_{2} = 0.284W$<br /><br />Therefore, the total power used in the circuit is 0.811W, and the power used in each resistor is 0.246W and 0.284W, respectively.<br /><br />d. Does the sum of the power used in each resistor in the circuit equal the total power used in the circuit? Explain.<br /><br />Yes, the sum of the power used in each resistor in the circuit equals the total power used in the circuit.<br /><br />This is because the power used in each resistor is a component of the total power used in the circuit. The total power used in the circuit is the sum of the power used in each resistor.<br /><br />In this case, the sum of the power used in each resistor is:<br />$P_{\text{total}} = P_{1} + P_{2}$<br />$0.811W = 0.246W + 0.284W$<br /><br />Therefore, the sum of the power used in each resistor in the circuit equals the total power used in the circuit.