In a laboratory experiment to verify the farrier these 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 

To solve this problem, we need to calculate the **magnetic flux** and the **average current** in the circuit based on the given information. Let's break this down step by step:

### 1. Understanding the given dataTo solve this problem, we need to calculate the **magnetic flux** and the **average current** in the circuit based on the given information. Let's break this down step by step:

### 1. Understanding the given 
- Number of turns of the coil, \( N = 100 \)
- Resistance of the coil, \( R_{\text{coil}} = 100 \, \Omega \)
- Resistance of the galvanometer, \( R_{\text{gal}} = 500 \, \Omega \)
- Total resistance in the circuit, \( R_{\text{total}} = R_{\text{coil}} + R_{\text{gal}} = 100 + 500 = 600 \, \Omega \)
- The induced EMF \( e(t) = 2t + 6 \, \text{V} \) (where \( t \) is in seconds)
- Time interval: \( 5 \, \text{s} \leq t \leq 10 \, \text{s} \)

### 2. Magnetic flux (\( \Phi \)):
Faraday's Law of Electromagnetic Induction states that the induced EMF is proportional to the rate of change of magnetic flux. Mathematically, this is expressed as:

\[
e(t) = -N \frac{d\Phi(t)}{dt}
\]

Here, \( e(t) = 2t + 6 \), so we can integrate to find the magnetic flux \( \Phi(t) \).

\[
\frac{d\Phi(t)}{dt} = -\frac{e(t)}{N}
\]

Substitute \( e(t) = 2t + 6 \) and \( N = 100 \):

\[
\frac{d\Phi(t)}{dt} = -\frac{2t + 6}{100}
\]

Integrating this with respect to \( t \) will give us the magnetic flux \( \Phi(t) \). Let's compute this.

### 3. Average current in the circuit:
The current in the circuit can be found using Ohm's Law:

\[
I(t) = \frac{e(t)}{R_{\text{total}}}
\]

Where \( R_{\text{total}} = 600 \, \Omega \).

The average current over the time interval \( 5 \leq t \leq 10 \) seconds can be found as:

\[
I_{\text{avg}} = \frac{1}{\Delta t} \int_{5}^{10} I(t) \, dt
\]

Where \( \Delta t = 10 - 5 = 5 \) seconds. Let's calculate both the magnetic flux and the average current.### 1. Magnetic Flux:
The magnetic flux \( \Phi(t) \) is given by:

\[
\Phi(t) = -\frac{t^2}{100} - \frac{3t}{50} \, \text{Wb} \quad (\text{Weber})
\]

### 2. Average Current:
The average current over the time interval \( 5 \leq t \leq 10 \) seconds is:

\[
I_{\text{avg}} = 0.035 \, \text{A} \quad (\text{35 mA})
\]

This is the average current flowing through the circuit during the experiment.

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Would you like more details or have any questions about the calculations? Here are five related questions for further exploration:

1. How does increasing the number of turns in the coil affect the induced EMF?
2. What role does the resistance of the galvanometer play in the total circuit resistance?
3. How can we determine the instantaneous current at any specific time within the interval?
4. Can you explain how the shape of the \( e(t) \) function influences the magnetic flux over time?
5. How would the average current change if the resistance of the coil was reduced?

**Tip:** When dealing with electromagnetic induction, the rate of change of magnetic flux is crucial in determining the induced EMF.

In a laboratory experiment to verify Faraday's law of electromagnetic induction, a coil of 100 turns having a resistance of 100 ohms is connected in series 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 the time interval 5 < t < 10 seconds. Find the magnetic flux and average current in the circuit.

Explore how to calculate the magnetic flux and average current in a circuit with a coil of 100 turns and resistance of 600 ohms. Using Faraday's law of electromagnetic induction and Ohm's law, this detailed solution is perfect for undergraduate students in physics or electrical engineering.

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