To analyze the statements, we can use Ohm's Law, which states that:
\[ V = I \times R \]
where:
- \( V \) is voltage,
- \( I \) is current, and
- \( R \) is resistance.
We have:
- Circuit A: \( V_A = 10 , \text{V} \)
- Circuit B: \( V_B = 15 , \text{V} \)
Now let's evaluate each statement.
A: If the resistance in circuit A equals the resistance in circuit B, then the current in circuit A is greater than the current in circuit B.
- False: If \( R_A = R_B \), then using \( I_A = \frac{V_A}{R_A} \) and \( I_B = \frac{V_B}{R_B} \), we see that \( I_A = \frac{10}{R} \) and \( I_B = \frac{15}{R} \). Thus, \( I_A < I_B \) because \( 10 < 15 \).
B: If the resistance in circuit A is greater than the resistance in circuit B, then the current in circuit A is better than the current in circuit B.
- False: If \( R_A > R_B \), \( I_A = \frac{10}{R_A} \) and \( I_B = \frac{15}{R_B} \). Given that \( R_A > R_B \), \( I_A \) can still be less than \( I_B \), so we cannot say that current in circuit A is "better" without clarity on the context of "better."
C: If the current in circuit A is greater than the current in circuit B, then the resistance in circuit A is less than the resistance in circuit B.
- True: If \( I_A > I_B \), then from Ohm's Law, we can say \( \frac{10}{R_A} > \frac{15}{R_B} \). This implies that \( R_A < R_B \).
D: If the current in circuit A equals the current in circuit B, then the resistance in circuit A equals the resistance in circuit B.
- False: If \( I_A = I_B \), it could be possible with different resistances in circuits A and B due to the different voltages. For instance, if \( R_A \) is different from \( R_B \), they may still have the same current depending on their resistances.
Therefore, the correct statement is:
C: If the current in circuit A is greater than the current in circuit B, then the resistance in circuit A is less than the resistance in circuit B.
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