In an electromagnetic wave, `E=1.2 sin (2xx 10^(6) t-k x) N/C` If the value of intensity of magnetic field is `(1)/(x pi)((A) (m)^(-1))`, then find `x`.

Answer

To solve the problem, we need to find the value of \( x \) in the expression for the magnetic field intensity given the electric field of an electromagnetic wave.

### Step 1: Understand the given electric field equation
The electric field \( E \) of the electromagnetic wave is given as:
\[ E = 1.2 \sin(2 \times 10^6 t - kx) \, \text{N/C} \]
Here, \( 1.2 \) is the amplitude of the electric field, and \( 2 \times 10^6 \) is the angular frequency \( \omega \).

### Step 2: Relate electric field and magnetic field in an electromagnetic wave
In an electromagnetic wave, the relationship between the electric field \( E \) and the magnetic field \( B \) is given by:
\[ c = \frac{E}{B} \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, \text{m/s} \).

### Step 3: Calculate the amplitude of the magnetic field
From the relationship \( c = \frac{E}{B} \), we can express the magnetic field \( B \) as:
\[ B = \frac{E}{c} \]
Substituting the values:
\[ B = \frac{1.2}{3 \times 10^8} \, \text{T} \]

### Step 4: Convert the magnetic field to intensity
The intensity \( I \) of an electromagnetic wave can also be expressed in terms of the electric field \( E \) and the magnetic field \( B \):
\[ I = \frac{E^2}{Z_0} \]
where \( Z_0 \) is the impedance of free space, approximately \( 377 \, \Omega \).

### Step 5: Calculate the intensity using the electric field
Substituting \( E \):
\[ I = \frac{(1.2)^2}{377} \, \text{W/m}^2 \]

### Step 6: Relate the intensity to the magnetic field intensity
The intensity of the magnetic field is given as:
\[ I = \frac{B^2}{\mu_0} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 7: Set the two expressions for intensity equal to each other
Now we have two expressions for intensity:
\[ \frac{(1.2)^2}{377} = \frac{B^2}{\mu_0} \]
Substituting for \( B \):
\[ B = \frac{1.2}{3 \times 10^8} \]
Substituting this into the intensity equation gives:
\[ \frac{(1.2)^2}{377} = \frac{\left(\frac{1.2}{3 \times 10^8}\right)^2}{4\pi \times 10^{-7}} \]

### Step 8: Solve for \( x \)
Given that the intensity of the magnetic field is \( \frac{1}{x \pi} \) A/m, we can set:
\[ \frac{1}{x \pi} = B \]
Substituting the value of \( B \):
\[ \frac{1}{x \pi} = \frac{1.2}{3 \times 10^8} \]
Solving for \( x \):
\[ x = \frac{3 \times 10^8}{1.2 \pi} \]

### Step 9: Calculate the numerical value of \( x \)
Calculating the value:
\[ x \approx \frac{3 \times 10^8}{1.2 \times 3.14} \approx \frac{3 \times 10^8}{3.768} \approx 7.95 \times 10^7 \]

### Final Answer
Thus, the value of \( x \) is approximately \( 7.95 \times 10^7 \).

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