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Ohm's Law is a fundamental principle in electronics and electrical engineering. It states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance. The mathematical representation of Ohm's Law is:

• \( I = \frac{V}{R} \)

where:

• \( I \) is the electric current in amperes (A)
• \( V \) is the voltage in volts (V)
• \( R \) is the resistance in ohms (Ω)

This relationship helps to determine how much current will flow in a circuit for any given voltage and resistance.
In RL circuits, this law is crucial for calculating the final steady-state current. In our example, with a 14.0 V battery and a 1.20 kΩ resistor, the final current \( I_f \) is calculated by dividing the voltage by the resistance, giving us approximately 0.0117 A.

The time constant \( \tau_L \) of an RL circuit is an important factor that describes how quickly currents in the circuit rise or fall. It is defined as the time required for the current to reach about 63.2% of its final value. The formula for the time constant in an RL circuit is given by:

• \( \tau_L = \frac{L}{R} \)

where:

• \( L \) is the inductance in henrys (H)
• \( R \) is the resistance in ohms (Ω)

The time constant gives an indication of how fast the circuit responds to changes. For example, a short time constant means the circuit responds quickly, whereas a longer time constant shows slower response. In the exercise, the time constant \( \tau_L \) was calculated to be approximately \( 5.25 \times 10^{-9} \text{ s} \).

Inductance is a property of an electrical conductor by which a change in current through it induces an electromotive force (EMF) in both the conductor itself (self-inductance) and in any nearby conductors (mutual inductance). It is measured in henrys (H), and in our exercise, the solenoid had an inductance of 6.30 µH. Inductance aims to resist changes in electrical current through Lenz's Law, which posits that an induced EMF always acts in a direction to oppose the change in current that is causing it.
The ability of the solenoid to store energy in a magnetic field as current passes through it is a key aspect to consider in RL circuits. High inductance in a circuit means it can oppose changes in current more effectively, which affects how quickly the circuit can reach a new steady-state current.

Exponential decay in an RL circuit describes how the current changes over time. The equation representing the current \( I(t) \) at a time \( t \) is:

• \( I(t) = I_f \left(1 - e^{-t/\tau_L}\right) \)

where:

• \( I_f \) is the final steady-state current
• \( e \) is the base of the natural logarithm
• \( t \) is the time elapsed
• \( \tau_L \) is the RL circuit time constant

Exponential decay shows how the current starts near zero and approaches its final value over time. The beauty of exponential behavior in RL circuits is that the rate of change itself changes over time, slowing as it reaches its final value. The exercise illustrates this as the current reaches 80% of its final value in about \( 8.42 \times 10^{-9} \text{ s} \).

Calculating the electric current in an RL circuit over time involves understanding how the circuit components interact. Initially, when a voltage is applied, the current is zero and increases according to the function:

• \( I(t) = I_f \left(1 - e^{-t/\tau_L}\right) \)

This gradual increase demonstrates how the inductor initially resists the change in current and then allows it to rise to the steady state.
At \( t = \tau_L \), a typical checkpoint in RL circuits, the current reaches \( I(\tau_L) = I_f \left(1 - \frac{1}{e}\right) \), which calculates to approximately 0.0074 A when using the figures provided in the exercise. This shows that even at the time constant, the current is still rising towards its final stable value of about 0.0117 A.