Final answer:
The inductance of the RL circuit is 0.100 H.
Step-by-step explanation:
The characteristic time constant (T) for an RL circuit is defined by the equation T = L/R, where L is the inductance and R is the resistance. Given that the time constant is 20.0 ns and the resistance is 5.00 MΩ, we can substitute these values into the equation and solve for L.
Rearranging the equation gives us L = T x R, so L = (20.0 ns) x (5.00 MΩ). Converting the time to seconds and the resistance to ohms, we get L = (20.0 x 10^-9 s) x (5.00 x 10^6 Ω) = 100 x 10^-3 H = 0.100 H.
The inductance L of an RL circuit with a characteristic time constant τ (tau) can be calculated using the formula τ = L/R, where R is the resistance.
For an RL circuit with a given time constant of 20.0 ns and a resistance of 5.00 MΩ, we can find the inductance by rearranging the formula to L = τ * R. Plugging in the values, we get L = 20.0 x 10-9 s * 5.00 x 106 Ω, which calculates to an inductance of 0.1 H.
When it comes to finding the resistance needed for a 1.00 ns time constant for rapid response in an oscilloscope, we would use the same formula but solve for R instead: R = τ / L.
Assuming the same inductance, we would rearrange to find the new resistance value for the desired quicker time constant.