Question

A $100.-V$ battery is connected in series with a $500.-\mathrm{\Omega }$ resistor. According to Faraday's Law of Induction, current can never change instantaneously, so there is always some "stray" inductance. Suppose the stray inductance is $0.200\mu \mathrm{H}$. How long will it take the current to build up to within $0.500\mathrm{\%}$ of its final value of $0.200\mathrm{A}$ after the resistor is con- nected to the battery?

Short answer

Answer: It will take approximately 2.99 ns for the current to build up to within 0.500% of its final value of 0.200 A in the given circuit.

Step-by-step solution

Determine the final value of the current

Since we are given the resistance (500 Ω) and the voltage (100 V) in the circuit, we can use Ohm's Law I = V/R to find the final value of current. Here, I = current, V = voltage, and R = resistance. I = 100V / 500Ω = 0.200 A The final value of the current in the circuit is 0.200 A.

Determine the time constant of the circuit

To find the time it takes for the current to build up, we need to know the time constant of the circuit. The time constant (τ) for an inductor-resistor circuit (RL circuit) is given by τ = L/R, where L is the inductance and R is the resistance. Given the stray inductance (0.200 μH = 0.200 x 10^{-6} H) and the resistance (500 Ω), we can find the time constant: τ = (0.200 x 10^{-6} H) / 500Ω ≈ 4 x 10^{-10} s

Find the time at which the current reaches 0.500% of its final value

The equation for the current buildup in an RL circuit is given by: I(t) = I_final * (1 - e^{-t/τ}) Where I(t) is the current at time t, I_final is the final value of the current, e is the base of the natural logarithm, and τ is the time constant. We need to find the time at which the current reaches 0.500% (0.995 times) of its final value. We can set up the equation: 0.995 * I_final = I_final * (1 - e^{-t/τ}) Divide both sides by I_final: 0.995 = 1 - e^{-t/τ} Rearrange the equation to get: e^{-t/τ} = 0.005 Now, take the natural logarithm (ln) of both sides of the equation: -t/τ = ln(0.005) Now isolate t: t = -τ * ln(0.005) We now have an equation for the time at which the current reaches 0.500% of its final value. Plug in the time constant (τ ≈ 4 x 10^{-10} s) we found in Step 2 and calculate: t ≈ -(4 x 10^{-10}s) * ln(0.005) ≈ 2.99 x 10^{-9} s The current build-up time t is approximately 2.99 x 10^{-9} seconds or 2.99 ns.

Key concepts

Ohm's Law

Ohm's Law is a fundamental principle in the field of electronics that relates voltage, current, and resistance in an electrical circuit. It is expressed as $I=\frac{V}{R}$, where $I$ stands for current in amperes, $V$ represents voltage in volts, and $R$ is the resistance in ohms. This simple formula is crucial for understanding how electricity flows through a circuit.
Understanding Ohm's Law helps in several ways:

• It allows you to calculate the current flowing through a circuit if you know the voltage and resistance.
• This law is used to design circuits and ensure that they operate safely and efficiently.
• It's also essential for troubleshooting, as deviations from expected current values can indicate an issue within the circuit.

In the context of the exercise, Ohm's Law is used to determine that the final current in the circuit is $0.200A$, using the given voltage of $100V$ and resistance of $500\mathrm{\Omega }$. This lays the groundwork for understanding how the current builds up in an RL circuit.

Inductor-Resistor Circuit

An inductor-resistor (RL) circuit is a simple but important type of electrical circuit, especially in understanding how current changes over time. It consists of a resistor (R) and an inductor (L) connected in series or parallel. The presence of the inductor means that the current does not increase or decrease instantly, but gradually over time.
Characteristics of RL Circuit:

• The inductor resists changes in current due to its property of inductance, measured in henrys (H).
• Resistance limits the flow of current and is measured in ohms (Ω).
• An RL circuit can be used to filter signals, control timing, and smooth out currents in various electronic applications.

In the exercise provided, the inductor creates a time delay in reaching the circuit's steady state due to its stray inductance of $0.200\mu H$. The resistance is $500\mathrm{\Omega }$, which is used alongside the inductance to calculate the time constant of the circuit, providing insight into how fast the current builds up.

Current Build-up Time

Current build-up time in an RL circuit is a measure of how quickly the current reaches a certain percentage of its final value. This process is not instantaneous due to the inductor’s resistance to changes in current, making it crucial to calculate the time it takes for the current to stabilize.
Key Points on Current Build-up:

• The time constant $\tau =\frac{L}{R}$ determines the rate of current increase, where $L$ is inductance and $R$ is resistance.
• Typically within five time constants, the current will reach near its final value (over 99%).
• In the exercise, calculating the exact time when the current reaches $0.995$ of its final value helps understand transient responses in circuits.

The calculated time constant for the given circuit is approximately $4\times {10}^{-10}s$. Using this, we find the build-up time to be approximately $2.99ns$. This short duration reflects the swift response of circuits with low inductance and high resistance.

Faraday's Law of Induction

Faraday's Law of Induction highlights the principle that a change in magnetic field within a closed loop induces an electromotive force (emf). This is foundational for understanding how inductors work in circuits. The law states that the induced emf in a circuit is proportional to the rate of change of the magnetic flux through the circuit.
Applications of Faraday’s Law:

• It helps explain how transformers, inductors, and many electrical machines work by converting mechanical energy to electrical energy and vice versa.
• Understanding Faraday's Law is vital for designing circuits that need to manage rapid current changes.
• In practical circuits, changes in current due to inductors follow an exponential curve as predicted by this law.

In the exercise, Faraday's Law explains why the current cannot change instantly but builds up over time. The stray inductance noted leads to a gradual increase of current to its final value, confirming the natural phenomenon predicted by Faraday's insights.