The equation of an equipotential line in an electric field is ` y = 2 x `, then electric field strength vectro at `1, 2` may be .

Options

A. `4hat(i)+3hat(j)`
B. `4hat(i)+8hat(j)`
C. `8hat(i)+4hat(j)`
D. `-8hat(i)+4hat(j)`

Explanation

To find the electric field strength vector at the point (1, 2) given that the equation of an equipotential line is \( y = 2x \), we can follow these steps:

### Step 1: Identify the slope of the equipotential line
The equation of the equipotential line is given as \( y = 2x \). The slope \( m_1 \) of this line can be identified directly from the equation.

\[ m_1 = 2 \]

### Step 2: Determine the slope of the electric field line
The electric field lines are perpendicular to the equipotential lines. Therefore, the product of their slopes \( m_1 \) and \( m_2 \) (the slope of the electric field line) must equal -1.

\[ m_1 \cdot m_2 = -1 \]

Substituting the value of \( m_1 \):

\[ 2 \cdot m_2 = -1 \]

Now, solve for \( m_2 \):

\[ m_2 = -\frac{1}{2} \]

### Step 3: Express the electric field vector in terms of its components
The slope \( m_2 \) can be expressed in terms of the components of the electric field vector. The electric field vector \( \vec{E} \) can be represented as:

\[ \vec{E} = E_x \hat{i} + E_y \hat{j} \]

The slope is given by the ratio of the \( y \) component to the \( x \) component:

\[ \frac{E_y}{E_x} = m_2 \]

Substituting the value of \( m_2 \):

\[ \frac{E_y}{E_x} = -\frac{1}{2} \]

### Step 4: Choose suitable values for \( E_x \) and \( E_y \)
From the ratio \( \frac{E_y}{E_x} = -\frac{1}{2} \), we can express \( E_y \) in terms of \( E_x \):

\[ E_y = -\frac{1}{2} E_x \]

Now, we can choose a convenient value for \( E_x \). Let's say \( E_x = 2 \). Then:

\[ E_y = -\frac{1}{2} \cdot 2 = -1 \]

### Step 5: Write the electric field vector
Thus, the electric field vector at the point (1, 2) can be expressed as:

\[ \vec{E} = 2 \hat{i} - 1 \hat{j} \]

### Conclusion
The electric field strength vector at the point (1, 2) is:

\[ \vec{E} = 2 \hat{i} - 1 \hat{j} \]