What should be the slope of the graph of the time constant vs. resistance? What should be the slope of the graph of the time constant vs. capacitance?
The slope of the graph of the time constant vs. resistance should be 1/C, where C is the capacitance. This is because the time constant (τ) of an RC circuit is equal to the product of the resistance (R) and the capacitance (C), τ = R * C. By rearranging the equation and solving for R, we get R = τ/C, which shows that resistance is inversely proportional to capacitance.
Similarly, the slope of the graph of the time constant vs. capacitance should be 1/R, where R is the resistance. This is because, using the same equation as above, when we solve for C, we get C = τ/R, showing that capacitance is inversely proportional to resistance.
To determine the slope of the graph of the time constant vs. resistance and the time constant vs. capacitance, we need to understand the relationship between these variables.
In an RC circuit, the time constant (τ) is given by the product of resistance (R) and capacitance (C), expressed as τ = RC. This formula indicates that the time constant is directly proportional to both resistance and capacitance.
1. Slope of the graph of time constant vs. resistance:
Since τ = RC, we can rewrite the formula as τ = kR, where k is a constant factor determined by the capacitance value. By comparing this equation with the slope-intercept form (y = mx + c), we can see that the slope (m) of the graph of time constant vs. resistance is equal to k, which represents the constant factor determined by capacitance.
2. Slope of the graph of time constant vs. capacitance:
Similarly, using the formula τ = RC, we can rewrite it as τ = (1/C)C, where 1/C represents the constant factor determined by resistance. Again, by comparing this equation with y = mx + c, we observe that the slope (m) of the graph of time constant vs. capacitance is equal to 1/C, which represents the constant factor determined by resistance.
Therefore, the slope of the graph of time constant vs. resistance is determined by the capacitance, whereas the slope of the graph of time constant vs. capacitance is determined by the resistance.