Problem 31 - Basic Technical Mathematics with Calculus

Question

An inductor in an electric circuit is essentially a coil of wire in which the voltage is affected by a changing current. By definition, the voltage caused by the changing current is given by \(V_{L}=L(d i / d t)\) where \(L\) is the inductance (in \(\mathrm{H}\) ). If \(V_{L}=12.0-0.2 t\) for a \(3.0-\mathrm{H}\) inductor, find the current in the circuit after 20 s if the initial current was zero.

Step-by-Step Solution

Answer

The current after 20 seconds is 66.66 A.

Step 1: Understand the given formula

The given problem involves the formula for voltage across an inductor: \( V_{L} = L \frac{di}{dt} \), where \( V_{L} \) is the voltage, \( L \) is the inductance (3.0 H in this case), and \( \frac{di}{dt} \) is the rate of change of current with respect to time.

Step 2: Substitute the given voltage expression

The voltage across the inductor is given as \( V_{L} = 12.0 - 0.2t \). We substitute this expression into the inductor equation: \( 12.0 - 0.2t = 3.0 \frac{di}{dt} \).

Step 3: Solve for \( \frac{di}{dt} \)

Rearrange the equation from Step 2 to express \( \frac{di}{dt} \):\[\frac{di}{dt} = \frac{12.0 - 0.2t}{3.0}\]= \( 4.0 - 0.0667t \).

Step 4: Integrate to find the current \( i(t) \)

Integrate \( \frac{di}{dt} = 4.0 - 0.0667t \) with respect to \( t \) to find the expression for \( i(t) \):\[i(t) = \int (4.0 - 0.0667t) \, dt = 4.0t - 0.0667 \frac{t^2}{2} + C\]= \( 4.0t - 0.03335t^2 + C \).

Step 5: Use initial conditions to find constant C

Since the initial current \( i(0) = 0 \), substitute into the expression: \( 0 = 4.0 \times 0 - 0.03335 \times 0^2 + C \), which gives \( C = 0 \).

Step 6: Find the current when \( t = 20 \) sec

Now, substitute \( t = 20 \) seconds into the expression for \( i(t) \) to find the current:\[i(20) = 4.0(20) - 0.03335(20)^2\]= \( 80 - 0.03335 \times 400 \)= \( 80 - 13.34 \) = 66.66 A.

Key Concepts

Electric Circuits

Electric circuits are pathways engineered to conduct electricity through them, utilizing various components such as resistors, batteries, capacitors, and inductors. Each of these components plays a unique role in the circuit.

Inductors, made by coiling wire, are crucial in circuits for managing current changes. They store energy in a magnetic field when electricity passes through them.

When the current in the circuit varies, the inductor reacts by inducing voltage across it. This can influence the behavior of the entire circuit, affecting how other components operate as well. Proper understanding of electric circuits is fundamental in crafting reliable and efficient electronic systems.

Voltage

Voltage, often referred to as electrical potential difference, is a crucial concept in electric circuits. It is the force that pushes electric current through a circuit, much like pressure pushing water through a pipe.

In the context of an inductor, the voltage induced across it is directly tied to how the current changes over time. This relationship is defined by the formula

• \( V_{L} = L \frac{di}{dt} \)

,where \( V_{L} \) represents the induced voltage, \( L \) is the inductance, and \( \frac{di}{dt} \) is the rate of change of current.

This voltage can either support or oppose the current flow, depending on the direction of the current change. Understanding voltage's role in circuits is essential for controlling how electronic devices function.

Rate of Change

The rate of change is a measure of how a quantity changes over time. In electric circuits, this is often applied to the current flowing through an inductor.

Captured mathematically by the expression \( \frac{di}{dt} \), it signifies the speed at which the current varies with time. This is significant because the rate of change of the current directly influences the voltage induced across the inductor.

This property helps engineers and scientists predict circuit behavior under varying conditions. It can indicate how quickly a system can adapt to changes and maintain its function.

Understanding the rate of change is a key part of mastering electronics and ensuring that circuits behave as intended when subjected to changing currents.

Integration in Calculus

Integration in calculus is a powerful mathematical tool used to find overall change, accumulation, or total value by combining rates of change. In electric circuits, it's often employed to determine the current based on the voltage across an inductor.

In this context, to find the total current \( i(t) \) from the differential expression \( \frac{di}{dt} \), you perform integration over time:

• \[i(t) = \int (4.0 - 0.0667t) \, dt\]

This integral helps determine how the current builds up in the circuit over time given a certain rate of change. Integration effectively gives us the "big picture" of the current flow from the rate of its change.

This process is critical in predicting and analyzing the total current in scenarios where it changes over time, making it a vital aspect of engineering and physics.