Problem 21 - Technical Mathematics with Calculus

Question

When the switch is closed, the current \(i\) will grow exponentially according to the following equation:$$\begin{array}{|lccccc|}\hline{2}{*} {\begin{array}{l}\text { Inductor Current } \\\\\text { When Charging }\end{array}} & {2}{|c} {i=\frac{E}{R}\left(1-e^{-R t / L}\right)} &{2}{c|} {\text { amperes }} & 1087 \\\\\hline\end{array}$$,Where \(L\) is the inductance in henries and \(R\) is the resistance in ohms. Find the current at 0.0750 s if \(R=6.25 \Omega, L=186 \mathrm{H},\) and \(E=249 \mathrm{V}\).

Step-by-Step Solution

Answer

Current \(i\) at \(t = 0.0750\) seconds is approximately 9.499 Amperes.

Step 1: Understand the given information

Here, we have an equation describing the current through an inductor when charging: \(i = \frac{E}{R}(1 - e^{-\frac{Rt}{L}})\), where \(R\) is the resistance, \(L\) is the inductance, \(E\) is the voltage and \(t\) is the time. We are given \(R = 6.25 \Omega\), \(L = 186 H\), and \(E = 249 V\). We need to find the current at \(t = 0.0750\) seconds.

Step 2: Plug in the given values

Substitute the given values for \(R\), \(L\), \(E\), and \(t\) into the equation: \(i = \frac{249}{6.25}(1 -e^{-\frac{6.25 \times 0.0750}{186}})\).

Step 3: Calculate the exponent

First, calculate the exponent value by dividing the product of resistance and time by the inductance: \(-\frac{6.25 \times 0.0750}{186} = -0.0025\).

Step 4: Evaluate the exponential expression

Now calculate the exponential term \(e^{-0.0025}\).

Step 5: Compute the current

Finally, compute the current \(i\) by evaluating the full expression: \(i = \frac{249}{6.25}(1 - e^{-0.0025})\). Multiply the result of \(1 - e^{-0.0025}\) with the fraction \(\frac{249}{6.25}\) to get the current \(i\) in amperes.

Key Concepts

Exponential Growth

Understanding exponential growth is crucial when analyzing inductor current in electrical circuits. In the case of an inductor, whenever a voltage is suddenly applied across it, as in this exercise's scenario, the current does not immediately reach its maximum value. Instead, it increases over time in a manner that can be described by an exponential function.

This exponential behavior is represented by the term \(1 - e^{-Rt/L}\) in the mathematical expression given for the current. The expression reflects how the current grows as time \(t\) progresses. At the very beginning, when \(t=0\), the exponent is zero leading to \(e^{0}=1\), which makes the term \(1 - e^{-Rt/L}\) equal to zero and hence indicates that the current starts at zero. As \(t\) increases, the exponent becomes more negative, making the value of \(e^{-Rt/L}\) smaller and the overall expression \(1 - e^{-Rt/L}\) larger, indicating an increase in current.

To illustrate this more clearly: as time passes, the effect of the exponential term diminishes, and the current value approaches the maximum steady state value of \(E/R\), which is dictated by Ohm's law. This is characteristic of exponential growth, where values increase rapidly at first and then more slowly as they approach a limiting value, known in this context as the final steady-state current.

Electrical Resistance

Electrical resistance, denoted as \(R\) and measured in ohms (\(\Omega\)), is a concept used to quantify how much an object opposes the flow of electric current. In our scenario, a resistor with a value of 6.25 ohms is part of a circuit that includes an inductor. The resistance plays a key role in how quickly the current reaches its steady-state value when charging the inductor.

Ohm's law, \(V=IR\), where \(V\) is voltage, \(I\) is current, and \(R\) is resistance, is fundamental in the understanding of electrical circuits. In the given exercise, we set up the maximum current that can be achieved in the circuit as \(E/R\), where \(E\) is the electromotive force or voltage. As the resistance increases, the maximum possible current decreases, and the time required for the current to reach a substantial percentage of its maximum value increases; this is due to the resistor slowing down the rate at which charge flows. This concept is key to grasping how current behaves over time in a resistive-inductive circuit.

Inductance

Inductance, represented by \(L\), is a measure of an inductor's ability to store energy in the magnetic field induced by the electric current passing through it. This property is essential for understanding the behavior of inductors in circuits. The unit of inductance is the henry (H), and in this particular exercise, we have an inductance value of 186 henries, which is quite substantial.

The role of inductance in the equation \(i = \frac{E}{R}(1 - e^{-Rt/L})\) is to determine the rate of current change in relation to time. A larger inductance implies a stronger magnetic field and hence a greater opposition to changes in current. This means that with a higher \(L\), the rate at which current rises towards the steady state is slower, as implied by the \(L\) in the denominator of the exponential's exponent \(e^{-Rt/L}\).

Therefore, the inductance not only affects the maximum current that the circuit will eventually reach but also significantly influences the time constant \(\tau = L/R\), which defines how fast the current will reach a certain percentage of its maximum value. Understanding how inductance impacts the change in current over time is crucial for solving problems involving inductive circuits and understanding how they store and release energy.