91Ó°ÊÓ

The concept of equilibrium current is crucial when dealing with circuits involving inductors. When a circuit reaches an equilibrium, it means the system has settled into a stable state where currents and voltages are consistent over time. In the context of an inductor connected to a battery, initially, there is no current because the inductor resists changes in current. Over time, the inductor allows the current to increase. Eventually, the effect of the inductor fades as if it has become just a regular piece of wire.

At equilibrium, the inductor doesn't oppose the change anymore and behaves only like a resistor with resistance, letting the battery's voltage fully drop across the resistor. This is when Ohm's Law becomes helpful.

• Voltage across inductor at equilibrium: Equivalent to zero
• Voltage across resistor: Equals the battery's emf

Solving for the equilibrium current using Ohm's Law, we find that the equilibrium current is given by:
\[ I = \frac{\mathscr{E}}{R} \]
Plugging in the values, \( \mathscr{E} = 100 \text{ V} \) and \( R = 10 \Omega \), results in an equilibrium current of 10 A.

Ohm's Law is a foundational principle in the study of electrical circuits, establishing the simple relationship between voltage, current, and resistance. According to Ohm's Law, the current (\( I \)) flowing through a conductor between two points is directly proportional to the voltage (\( V \)) across the two points and inversely proportional to the resistance (\( R \)) of the conductor. The law is mathematically expressed as:
\[ V = IR \]
In the case of our coil, when the circuit reaches equilibrium, Ohm's Law helps determine the current flowing through the resistor part of the resistor-inductor series.

This simple relationship is instrumental in calculating the current because it allows us to find the current by rearranging the equation to:

• \( I = \frac{V}{R} \)

By applying Ohm's Law in circuits with resistors and obtaining the equilibrium current in scenarios where an inductor is present, we can predict the behavior of various electrical components accurately in a circuit.

In circuits containing inductors, an important aspect is the energy stored in the magnetic field. When an electric current passes through an inductor, it creates a magnetic field around it. The energy stored in this magnetic field is a critical feature of inductors, particularly in applications involving magnetic energy storage and electromagnetic devices. The formula used to determine the energy stored in the magnetic field within an inductor is:
\[ U = \frac{1}{2}LI^2 \]
Where \( U \) is the energy, \( L \) is the inductance, and \( I \) is the current.

Applying this to the given problem:

• Inductance \( L = 2.0 \text{ H} \)
• Current \( I = 10 \text{ A} \)

We calculate the energy to be:
\[ U = \frac{1}{2} \times 2.0 \times 10^2 \]
This results in the energy stored as \( 100 \text{ J} \). Such energy is harnessed in various applications like transformers and inductive kickbacks, showcasing the versatility and usefulness of magnetic fields in energetic systems.