Question

Your \(RL\) circuit has a characteristic time constant of \(20.0{\rm{ }}ns\), and a resistance of \(5.00{\rm{ }}M\Omega \). (a) What is the inductance of the circuit? (b) What resistance would give you a \(1.00{\rm{ }}ns\) time constant, perhaps needed for quick response in an oscilloscope?

Short answer

a.The inductance of the circuit\(0.1H\).

b. The \(100{\rm{ }}M\Omega \) resistance would give you a \(1.00{\rm{ }}ns\) time constant

Step-by-step solution

Concept Introduction

Resistance is a measurement of the resistance to current flow in an electrical circuit. Resistance in ohms is denoted by the Greek letter omega\(\left( \Omega \right)\). Ohms are named after Georg Simon Ohm\(\left( {1784 - 1854} \right)\), a German scientist who studied the relationship between voltage, current, and resistance.

Information Provided

• Time constant of the circuit:\(20.0{\rm{ }}ns\)
• Resistance value for the circuit: \(5.00{\rm{ }}M\Omega \)

Calculating Inductance of the Circuit

a)

In\(RL\)circuits, the time constant,\(\tau \),isgivenas

\(\tau = \frac{L}{R}\)

Asaresult,wecanfortheinductanceas

\(L = \tau R\)

Numerically, in our case, we will have

\(\begin{array}{c}L = 2 \times {10^{ - 8}} \times 5 \times {10^6}\\ = 0.1H\end{array}\)

Therefore, the inductance value is\(0.1H\).

Calculating Resistance

b)

With the same inductance, we'll need a resistance \(20\) times larger to get a time constant \(20\) times smaller, which implies its value will be \(100{\rm{ }}M\Omega \).