In a laboratory experiment to verify the Faraday law of electromagnetic induction a coil of 100 turns having a resistance of 100 ohms is connected in series circuit with a galvanometer of 500 ohms resistance, the coil is placed in a magnetic field and the induced emf, e=2t +6 was recorded in time 5<t<10 seconds interval. Find the magnetic flux and average current in the circuit

2024-09-16 13:31:37



To solve this problem, we need to calculate both the magnetic flux \(\Phi\) and the average current \(I_{\text{avg}}\) in the circuit.

### Given:
- Number of turns of the coil \(N = 100\)
- Resistance of the coil \(R_{\text{coil}} = 100 \, \Omega\)
- Resistance of the galvanometer \(R_{\text{galv}} = 500 \, \Omega\)
- Total resistance \(R = R_{\text{coil}} + R_{\text{galv}} = 100 + 500 = 600 \, \Omega\)
- The induced EMF is given by \(e(t) = 2t + 6\) in the time interval \(5 < t < 10 \, \text{seconds}\)

### Step 1: Magnetic flux
Faraday's Law of Electromagnetic Induction states that the induced EMF is related to the rate of change of magnetic flux through the coil:
\[
e(t) = -N \frac{d\Phi}{dt}
\]
Rearranging for magnetic flux \(\Phi\):
\[
\frac{d\Phi}{dt} = -\frac{e(t)}{N}
\]
Substitute \(e(t) = 2t + 6\) and \(N = 100\):
\[
\frac{d\Phi}{dt} = -\frac{2t + 6}{100}
\]
Integrating both sides with respect to \(t\) over the interval \(5 \leq t \leq 10\), we can find the change in magnetic flux over that period.

### Step 2: Average current in the circuit
Ohm's law gives us the relation between current and EMF:
\[
I(t) = \frac{e(t)}{R}
\]
Substitute \(e(t) = 2t + 6\) and \(R = 600 \, \Omega\):
\[
I(t) = \frac{2t + 6}{600}
\]
The average current over the interval \(5 \leq t \leq 10\) is:
\[
I_{\text{avg}} = \frac{1}{\Delta t} \int_5^{10} I(t) \, dt
\]
where \(\Delta t = 10 - 5 = 5 \, \text{seconds}\).

Let's compute both the magnetic flux and the average current.

### Results:
1. **Magnetic flux change (\(\Delta \Phi\)):**
   \[
   \Delta \Phi = -\frac{21}{20} \, \text{Wb} = -1.05 \, \text{Wb}
   \]
   This represents the total change in magnetic flux over the time interval from \(t = 5\) to \(t = 10\) seconds.

2. **Average current (\(I_{\text{avg}}\)):**
   \[
   I_{\text{avg}} = 0.035 \, \text{A} = 35 \, \text{mA}
   \]
   The average current in the circuit during the time interval is 35 mA.

Let me know if you want further details, or have any questions. Here are five related questions:

1. How is the magnetic flux through a coil related to the induced EMF?
2. Why is the resistance of the galvanometer added to the total resistance of the circuit?
3. How would the current change if the resistance of the coil was doubled?
4. What is the significance of the negative sign in the magnetic flux change?
5. How would the experiment change if the coil had fewer turns?

**Tip:** To minimize resistance in circuits with galvanometers, always account for the internal resistance of both the coil and the galvanometer.

In a laboratory experiment to verify the Faraday law of electromagnetic induction a coil of 100 turns having a resistance of 100 ohms is connected in series circuit with a galvanometer of 500 ohms resistance, the coil is placed in a magnetic field and the induced emf, e=2t +6 was recorded in time 5<t<10 seconds interval. Find the magnetic flux and average current in the circuit.

This problem involves verifying Faraday's law of electromagnetic induction for a coil of 100 turns and calculating the magnetic flux and average current in a circuit. Explore key concepts in electromagnetism and Ohm's law using step-by-step solutions for induced EMF and current over a time interval of 5 to 10 seconds.

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