A metal dise of radins `0.1m` spins about a horizontal axis lyinig in the magnetic meridian' at a speed of 5 revis. If the horizontal componeat of the earth's.field is `B=2 xx 10^(-5) Wb / m^2`, the potential difference between the centre and the outer edge of the disc is `zxx 10^(-6) V` Find `z`.

Answer

To solve the problem, we will calculate the potential difference (EMF) generated between the center and the outer edge of a spinning metal disk in a magnetic field. Here are the step-by-step calculations:

### Step 1: Understand the given parameters
- Radius of the disk (R) = 0.1 m
- Speed of the disk (f) = 5 revolutions per second
- Horizontal component of the Earth's magnetic field (B) = 2 × 10^(-5) Wb/m²

### Step 2: Convert the frequency to angular velocity
The angular velocity (ω) in radians per second can be calculated using the formula:
\[ \omega = 2\pi f \]
Substituting the given frequency:
\[ \omega = 2\pi \times 5 = 10\pi \, \text{rad/s} \]

### Step 3: Use the formula for EMF
The formula for the electromotive force (EMF) induced in a rotating disk is given by:
\[ E = \frac{1}{2} B \omega R^2 \]
Substituting the values we have:
\[ E = \frac{1}{2} \times (2 \times 10^{-5}) \times (10\pi) \times (0.1)^2 \]

### Step 4: Calculate the EMF
Now, let's calculate the EMF step by step:
1. Calculate \(R^2\):
\[ R^2 = (0.1)^2 = 0.01 \, \text{m}^2 \]
2. Substitute \(R^2\) into the EMF formula:
\[ E = \frac{1}{2} \times (2 \times 10^{-5}) \times (10\pi) \times 0.01 \]
3. Simplify the expression:
\[ E = (10^{-5}) \times (10\pi) \times 0.01 = 10^{-5} \times 10\pi \times 10^{-2} \]
\[ E = \pi \times 10^{-6} \, \text{V} \]

### Step 5: Relate EMF to the potential difference
We are given that the potential difference between the center and the outer edge of the disk is \(z \times 10^{-6} \, \text{V}\). Therefore, we can set up the equation:
\[ z \times 10^{-6} = \pi \times 10^{-6} \]

### Step 6: Solve for \(z\)
By cancelling \(10^{-6}\) from both sides, we get:
\[ z = \pi \]

### Step 7: Approximate the value of \(z\)
Using the approximate value of \(\pi \approx 3.14\):
\[ z \approx 3.14 \]

Thus, the value of \(z\) is approximately \(3.14\).

### Final Answer
\[ z \approx 3.14 \]

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