A Coil Has 500 Turns And Self-inductance 7.50 MH.The Current In The Coil Varies With Time According Toi=(680mA)cos[t/(0.0250s)].Part

A Coil Has 500 Turns And Self-inductance 7.50 MH.The Current In The Coil Varies With Time According Toi=(680mA)cos[t/(0.0250s)].Part

Understanding the behavior of electrical coils, especially in the context of inductance and time-varying currents, is fundamental in electromagnetic theory and practical applications such as transformers, inductors, and electrical circuits. In this article, we explore the characteristics of a coil with specific parameters, analyze its inductance, current variation, and the resulting electromotive force (EMF), along with relevant calculations and implications.

Fundamental Concepts of Inductance and Coil Behavior

What Is Self-Inductance?

Self-inductance is the property of a coil or inductor that opposes changes in the current flowing through it. It is defined as the ratio of the magnetic flux linkage to the current causing it:

L = Φ / I

where:

• L is the inductance (measured in henries, H),
• Φ is the magnetic flux linkage (weber-turns),
• I is the current (amperes).

A higher inductance indicates a stronger opposition to changes in current, leading to more significant induced EMF when the current varies.

Parameters of the Given Coil

The specific parameters provided are:

• Number of turns, N = 500
• Self-inductance, L = 7.50 millihenries = 7.50 × 10⁻³ H
• Current variation with time: i(t) = (680 mA) cos [t / (0.0250 s)]

These parameters form the basis for analyzing the coil's behavior under the specified current variation.

Analysis of the Current Variation

Current as a Function of Time

The current in the coil is given by:

i(t) = 680 mA × cos [ t / 0.0250 s ]

Expressed in amperes:

i(t) = 0.680 A × cos [ t / 0.0250 s ]

This describes a sinusoidal current with an angular frequency:

ω = 1 / 0.0250 s = 40 rad/sec

which implies the period:

T = 2π / ω ≈ 2π / 40 ≈ 0.157 s.

Key observations:

• The current oscillates between +0.680 A and -0.680 A.


• The cosine function indicates a smooth, periodic change in current over time.

Implications of the Current Variation

Since the current varies sinusoidally, the coil experiences an alternating magnetic flux, which induces an EMF opposing the change in current — a fundamental principle of electromagnetic induction.

Calculating the Induced EMF

Faraday’s Law of Electromagnetic Induction

The induced EMF (voltage) in the coil is given by:

ε(t) = -L × (d i(t) / dt)

where:

• d i(t) / dt is the time derivative of the current.

Derivative of the Current

Given:

i(t) = 0.680 A × cos (ω t)

then:

d i(t) / dt = -0.680 A × ω × sin (ω t)

Substituting ω = 40 rad/sec:

d i(t) / dt = -0.680 A × 40 × sin (40 t) ≈ -27.2 A × sin (40 t)

Maximum Induced EMF

The maximum EMF occurs when sin (ω t) = ±1:

|ε_max| = L × ω × 0.680 A

Calculating:

|ε_max| = 7.50 × 10⁻³ H × 40 rad/sec × 0.680 A ≈ 0.205 V

Interpretation:

• The coil induces an EMF with a maximum magnitude of approximately 0.205 volts.


• The phase difference between current and induced EMF is 90°, typical of inductive circuits.

Power and Energy Considerations

Instantaneous Power

The instantaneous power delivered or absorbed by the coil:

p(t) = i(t) × ε(t)

Substituting:

p(t) = (0.680 cos ω t) × (-L ω) × (0.680 sin ω t)

which simplifies to:

p(t) = -L ω × (0.680)² × cos ω t × sin ω t

Using the identity:

cos ω t × sin ω t = (1/2) × sin 2 ω t

we get:

p(t) = - (1/2) L ω × (0.680)² × sin 2 ω t

Maximum power:

|p_max| = (1/2) × L × ω × (0.680)²

Calculating:

|p_max| = 0.5 × 7.50 × 10⁻³ H × 40 × (0.680)² ≈ 0.5 × 7.50 × 10⁻³ × 40 × 0.462 ≈ 0.055 W

This indicates the coil exchanges energy with the circuit periodically but does not dissipate power as heat, assuming ideal conditions.

Applications and Practical Implications

In Electrical Circuits

Understanding the behavior of coils with specified inductance and current variation is essential in designing:

• Transformers for voltage regulation
• Inductive filters in electronic circuits
• Wireless power transfer systems
• Inductive sensors and coils in measurement devices

In Signal Processing

The sinusoidal current and induced EMF are critical in AC circuit analysis, allowing engineers to design systems with predictable phase relationships and power flow characteristics.

In Power Systems

Large inductors are used to control current flow and smooth out fluctuations, essential for maintaining stability in power grids.

Additional Calculations and Considerations

Energy Stored in the Magnetic Field

The energy stored in the magnetic field of the inductor at any instant:

U = (1/2) L I²

At peak current I = 0.680 A:

U = 0.5 × 7.50 × 10⁻³ H × (0.680 A)² ≈ 0.5 × 7.50 × 10⁻³ × 0.462 ≈ 1.73 × 10⁻³ J

This energy is cyclically stored and released during each oscillation.

Effect of Resistance

While the problem assumes an ideal inductor, real coils have resistance which causes energy dissipation as heat. The presence of resistance affects:

• The amplitude of current oscillations
• The phase relationship
• The quality factor (Q) of the coil

If resistance R is significant, the circuit becomes a damped oscillator, and the maximum current and EMF would be lower.

Summary and Key Takeaways

• The coil's inductance of 7.50 mH opposes rapid changes in current, inducing an EMF proportional to the rate of change.
• The sinusoidal current with a maximum of 0.680 A oscillates with a period of approximately 0.157 seconds, driven by the angular frequency ω = 40 rad/sec.
• The maximum induced EMF in the coil is approximately 0.205 volts, opposing the change in current per Faraday’s law.
• Energy is cyclically stored in the magnetic field, with a maximum of around 1.73 millijoules during each oscillation.
• Understanding these parameters is crucial in designing and analyzing AC circuits, transformers, and inductive components.

In conclusion, analyzing a coil with specific parameters provides valuable insights into electromagnetic phenomena, enabling engineers and scientists to optimize the performance of electrical devices and systems that rely on inductance and time-varying currents.