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

In this problem, we are tasked with determining the magnetic flux and the average current in the circuit for a time interval \( 5 \, \text{seconds} < t < 10 \, \text{seconds} \). The induced electromotive force (emf) is given as \( e(t) = 2t + 6 \), and we know the resistance values of the circuit components. Here is how we proceed step by step:

### Given Information:
- Number of turns in the coil, \( N = 100 \)
- Coil resistance, \( R_{\text{coil}} = 100 \, \Omega \)
- Galvanometer resistance, \( R_{\text{galv}} = 500 \, \Omega \)
- Total resistance in the circuit, \( R = R_{\text{coil}} + R_{\text{galv}} = 100 + 500 = 600 \, \Omega \)
- Induced emf as a function of time, \( e(t) = 2t + 6 \)
- Time interval: \( t = 5 \, \text{seconds} \) to \( t = 10 \, \text{seconds} \)

### Step 1: Calculate the Magnetic Flux

Faraday's Law of Electromagnetic Induction states that:
\[
e(t) = -N \frac{d\Phi_B}{dt}
\]
Where:
- \( N \) is the number of turns in the coil.
- \( \Phi_B \) is the magnetic flux through the coil.

Rearranging the formula for magnetic flux:
\[
\frac{d\Phi_B}{dt} = -\frac{e(t)}{N}
\]
Substituting \( e(t) = 2t + 6 \) and \( N = 100 \):
\[
\frac{d\Phi_B}{dt} = -\frac{2t + 6}{100}
\]
Now, we can integrate this expression over time to find the magnetic flux \( \Phi_B \). The integral of \( \frac{d\Phi_B}{dt} \) will give us \( \Phi_B(t) \).

The initial flux at \( t = 5 \) seconds and the final flux at \( t = 10 \) seconds will be calculated from this expression.

### Step 2: Calculate the Average Current

The current in the circuit is related to the induced emf by Ohm’s Law:
\[
I(t) = \frac{e(t)}{R}
\]
Where:
- \( e(t) = 2t + 6 \)
- \( R = 600 \, \Omega \)

To find the average current over the time interval \( 5 \, \text{seconds} < t < 10 \, \text{seconds} \), we calculate:
\[
I_{\text{avg}} = \frac{1}{t_2 - t_1} \int_{t_1}^{t_2} I(t) \, dt
\]
Where:
- \( t_1 = 5 \, \text{seconds} \)
- \( t_2 = 10 \, \text{seconds} \)
- \( I(t) = \frac{2t + 6}{600} \)

We will now compute the results step by step.

Let's start with the magnetic flux calculation and then move on to the current.### Results:

1. **Magnetic Flux**:
   - Magnetic flux at \( t = 5 \) seconds: \( \Phi_5 = -\frac{11}{20} \, \text{Wb} = -0.55 \, \text{Wb} \)
   - Magnetic flux at \( t = 10 \) seconds: \( \Phi_10 = -\frac{8}{5} \, \text{Wb} = -1.6 \, \text{Wb} \)
   - Change in magnetic flux over the time interval: \( \Delta \Phi = -\frac{21}{20} \, \text{Wb} = -1.05 \, \text{Wb} \)

2. **Average Current**:
   - The average current in the circuit over the interval \( 5 \, \text{seconds} < t < 10 \, \text{seconds} \) is \( I_{\text{avg}} = 0.035 \, \text{A} \) or 35 mA.

### Summary:
- The magnetic flux changes by \( -1.05 \, \text{Wb} \) during the time interval.
- The average current induced in the circuit is 35 mA.

Would you like further details or have any questions?

### Relative Questions:
1. How does the number of turns in the coil affect the induced emf?
2. What is the significance of the negative sign in the change of magnetic flux?
3. How would increasing the resistance in the circuit impact the average current?
4. Can we use this method to find the instantaneous current at a specific time?
5. How does Faraday's Law of Induction relate to Lenz's Law?

### Tip:
Always remember that increasing resistance in the circuit will reduce the induced current for the same emf, as per Ohm’s Law.

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 magnetic flux and average current using Faraday's Law of Electromagnetic Induction in a coil with 100 turns. This problem involves a time-varying induced emf (e = 2t + 6) over a 5 to 10 second interval, with key calculations using Ohm's Law and integration.

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