The coil of a large electromagnet has an internal resistance of R = 8.50 R. It is in series with a source of EMF of 80.75 V and a switch. The switch is initially open. At ( t=0 \mathrm{~s} ), the switch is closed. The graph shows the electric current flowing through the coil as a function of time. What is the inductance of the coil? Please, notice that the curve passes through at least one grid intersection point.

Answer

🧠 RL Circuit Inductance Calculation — Detailed Explanation

In electrical engineering, analyzing how current behaves in circuits composed of resistors (R) and inductors (L) is crucial for designing reliable, safe, and efficient systems. An RL circuit—one containing a resistor and an inductor in series—offers insight into how current changes over time when a voltage source is applied. The resistor opposes the flow of current by converting electrical energy into heat, whereas the inductor opposes sudden changes in current by generating a counter voltage (EMF), which is proportional to the rate of change of current through it.

⚡ The Nature of RL Circuits

An RL circuit does not respond to a voltage source instantly. When the switch is closed and voltage is applied, current does not jump directly to its maximum possible value. This is due to the property of inductance: inductors resist changes in current flow. Initially, the inductor creates a back electromotive force (EMF) that opposes the incoming current. Over time, as the magnetic field around the coil builds up, the rate of change of current decreases and the back EMF falls. Eventually, the current settles at a final steady-state value determined by the resistance in the circuit, as if the inductor becomes a simple connecting wire.

📈 How Does the Current Rise?

The rise in current follows an exponential curve described mathematically by:

I(t) = Imax (1 - e-Rt/L)

This equation tells us how the current increases from zero toward its maximum value Imax over time t, depending on both the resistance R and inductance L. The term τ = L/R is called the time constant of the circuit. It characterizes how fast or slow the current approaches its maximum value:

• After one time constant (t = τ), the current reaches about 63% of its final value.
• After two time constants, it reaches about 86%.
• After five time constants, it’s more than 99% of the final value.

🧪 Purpose of This Experiment

This experiment seeks to determine the inductance L of a coil based on how the current increases over time. Instead of using physical instruments to directly measure inductance (which may be expensive or unavailable), we extract it from the exponential nature of the current curve.

📘 Given Data from the Experiment

• Resistance of the coil: R = 8.5 Ω
• Applied voltage (EMF): E = 80.75 V
• Initial condition: Switch is closed at t = 0
• From the current-time graph: I(t = 4s) = 5.0 A
• Final current: Imax = E / R = 80.75 / 8.5 = 9.5 A

📐 The Mathematical Model Used

I(t) = Imax (1 - e-Rt/L)

Where:

• I(t) = current at time t
• Imax = E/R
• R = resistance
• L = inductance (to solve for)
• t = time in seconds

🔍 Step-by-Step Calculations

Step 1: Plug values into the exponential current formula

5.0 = 9.5 (1 - e-4R/L)

Step 2: Divide both sides by 9.5

0.5263 = 1 - e-4R/L

Step 3: Rearranging

e-4R/L = 0.4737

Step 4: Take the natural logarithm of both sides

-4R/L = ln(0.4737)

Substitute R = 8.5:

-34.0 / L = -0.747

Step 5: Solve for L

L = 34.0 / 0.747 = 45.5 H

✅ Final Result

Therefore, the final answer is L ≈ 45.5H

🧲 Interpretation of the Result

This is a very large inductance. Inductors of this size are typically found in:

• Large industrial electromagnets
• Heavy-duty transformers
• Power conditioning equipment
• Pulse shaping circuits in laboratories

🛠️ Real-World Relevance

This method of determining inductance from a voltage–current–time relationship is common in both academic and professional electrical engineering practice.

📚 Summary Table

🧪 Educational Insight

🚀 Extended Learning Opportunities

• Vary R or L and observe time constant effects
• Analyze charging behavior in RC circuits
• Explore AC behavior using XL = 2πfL
• Calculate energy stored: E = ½LI²

📝 Conclusion

The RL circuit analysis demonstrated how engineers derive inductance values using simple graph data and exponential functions. The result of L ≈ 45.5 H confirms the practical utility of mathematical modeling in circuit design and verification.