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Chapter 21: Problem 10

A coil of 40 henry inductance is connected in series with a resistance of 8 ohm and the combination is joined to the terminals of a 2 volt battery. The time constant of the circuit is (a) 5 seconds (b) \(1 / 5\) seconds (c) 40 seconds (d) 20 seconds

Short Answer

Expert verified
The time constant of the circuit is 5 seconds (option a).

Step by step solution

01

Identify the Formula for Time Constant

In an electrical circuit that consists of an inductor and a resistor in series, the time constant \( \tau \) is given by the formula \( \tau = \frac{L}{R} \), where \( L \) is the inductance in henrys and \( R \) is the resistance in ohms.
02

Plug Values into the Formula

Using the formula from Step 1, substitute \( L = 40 \) henries and \( R = 8 \) ohms into \( \tau = \frac{L}{R} \). This gives \( \tau = \frac{40}{8} \).
03

Calculate the Time Constant

Perform the division to find the time constant: \( \tau = \frac{40}{8} = 5 \) seconds.
04

Select the Correct Answer

From the calculated result, the time constant \( \tau = 5 \) seconds. Comparing with the given options, option (a) 5 seconds is the correct answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inductance
Inductance is a fundamental concept in the realm of electromagnetic fields and circuits. Imagine a coil or spool of wire; when an electric current passes through it, it generates a magnetic field around itself. Inductance is a measure of how effectively a coil can store energy within that magnetic field. It is measured in a unit called henrys (H).

The key role of inductance in circuits is to resist changes in the current flowing through it. This property arises because any change in the current also changes the magnetic field, which induces a voltage opposing the change according to Lenz's Law.
  • Inductors store energy in the magnetic field they create.
  • High inductance means the coil strongly resists changes in current.
  • Common analogies compare inductors to flywheels which store mechanical energy.
Resistance
Resistance is the opposition to the flow of electric current through a conductor. Think of it like friction that resists movement; resistance resists the flow of electric charge. The unit of resistance is the ohm (惟). Resistance in a circuit is influenced by material, length, and cross-sectional area of the conductive path.

In our context, resistance is crucial because it affects how much current flows for a given voltage - according to Ohm's Law, which states:

\[ V = IR \]
Where \( V \) is the voltage, \( I \) is the current, and \( R \) is the resistance. This relationship tells us everything we need to know about how electricity behaves in a resistive component of the circuit.
  • Resistance causes electrical energy to be converted to heat.
  • Materials with high resistance are called insulators.
  • Copper is a common wire material due to its low resistance.
Electric Circuit Analysis
Electric circuit analysis is the examination of complex electrical systems to understand how electricity flows and how components like resistors and inductors influence this flow. In a combination of inductance and resistance, known as an RL circuit, understanding their interaction is critical.

The time constant is a crucial value in RL circuits, representing the time it takes for the current to change significantly鈥攕pecifically about 63.2% of its total change鈥攁fter a sudden change in voltage. The formula for the time constant \( \tau \) is:

\[ \tau = \frac{L}{R} \]
Here, \( L \) is inductance and \( R \) is resistance. This formula shows how the inductance and resistance together determine the speed of transient response in a circuit after a change.
  • Electric circuit analysis helps in designing and understanding electronic devices.
  • Time constant gives insight into how quickly circuits reach steady-state operations.
  • Knowing the time constant helps in predicting the behavior of circuits over time.

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Most popular questions from this chapter

In a circuit \(L, C\) and \(R\) are connected in series with an alternating voltage source of frequency \(f .\) The current leads the voltage by \(45^{\circ}\). The value of \(C\) is (a) \(\frac{1}{\pi f(2 \pi f L-R)}\) (b) \(\frac{1}{2 \pi f(2 \pi f L-R)}\) (c) \(\frac{1}{\pi f(2 \pi f L+R)}\) (d) \(\frac{1}{2 \pi f(2 \pi f L+R)}\)

A condenser of capacity \(C\) is charged to a potential difference of \(V_{1} .\) The plates of the condenser are then connected to an ideal inductor of inductance \(L .\) The current through the inductor when the potential difference across the condenser reduces to \(V_{2}\) is (a) \(\left(\frac{C\left(V_{1}-V_{2}\right)}{L}\right)^{\frac{1}{2}}\) (b) \(\frac{C\left(V_{1}^{2}-V_{2}^{2}\right)}{L}\) (c) \(\frac{C\left(V_{1}^{2}+V_{2}^{2}\right)}{L}\) (d) \(\left(\frac{C\left(V_{1}^{2}-V_{2}^{2}\right)}{L}\right)^{\frac{1}{2}}\)

A coil of 100 turns and area of cross-section \(0.001 \mathrm{~m}^{2}\) is free to rotate about an axis. The coil is placed perpendicular to a magnetic field of \(1 \mathrm{~Wb} \mathrm{~m}^{-2}\). If the coil is rotated rapidly through an angle of \(180^{\circ}\), how much charge will flow through the coil? The resistance of the coil is \(10 \Omega\). (a) \(0.01 \mathrm{C}\) (b) \(0.02 \mathrm{C}\) (c) \(0.04 \mathrm{C}\) (d) \(0.08 \mathrm{C}\)

A rectangular, a square, a circular and an elliptical loop, all in the \((x-y)\) plane, are moving out of a uniform magnetic field with a constant velocity, \(\vec{V}=v \hat{i}\). The magnetic field is directed along the negative \(z\) axis direction. The induced emf, during the passage of these loops, out of the field region, will not remain constant for (a) the circular and the elliptical loops (b) only the elliptical loop (c) any of the four loops (d) the rectangular, circular and elliptical loops

A conducting circular loop is placed in a uniform magnetic field \(0.04 \mathrm{~T}\) with its plane perpendicular to the magnetic field. The radius of the loop starts shrinking at \(2 \mathrm{~mm} / \mathrm{s}\). The induced emf in the loop when the radius is \(2 \mathrm{~cm}\) is (a) \(4.8 \pi \mu \mathrm{V}\) (b) \(0.8 \pi \mu \mathrm{V}\) (c) \(1.6 \pi \mu \mathrm{V}\) (d) \(3.2 \pi \mu \mathrm{V}\)
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