There is a uniform electric field of strength `10^(3)V//m` along `y`-axis. A body of mass `1g` and charge `10^(-6)C` is projected into the field from origin along the positive `x`-axis with a velocity `10 m//s`. Its speed in `m//s` after `10 s` is (Neglect gravitation)

Options

A. 10
B. `5sqrt(2)`
C. `10sqrt(2)`
D. 20

Explanation

To solve the problem, we will follow these steps:

### Step 1: Identify the Given Information
- Electric field strength, \( E = 10^3 \, \text{V/m} \)
- Mass of the body, \( m = 1 \, \text{g} = 1 \times 10^{-3} \, \text{kg} \)
- Charge of the body, \( q = 10^{-6} \, \text{C} \)
- Initial velocity along the x-axis, \( v_x = 10 \, \text{m/s} \)
- Time of projection, \( t = 10 \, \text{s} \)

### Step 2: Calculate the Force on the Charged Body
The force \( F \) acting on the charged body due to the electric field is given by:
\[ F = qE \]
Substituting the values:
\[ F = (10^{-6} \, \text{C})(10^3 \, \text{V/m}) = 10^{-3} \, \text{N} \]

### Step 3: Calculate the Acceleration of the Body
Using Newton's second law, the acceleration \( a \) can be calculated as:
\[ a = \frac{F}{m} \]
Substituting the values:
\[ a = \frac{10^{-3} \, \text{N}}{1 \times 10^{-3} \, \text{kg}} = 1 \, \text{m/s}^2 \]

### Step 4: Determine the Velocity in the y-direction After Time \( t \)
Since the initial velocity in the y-direction \( v_{y0} = 0 \), the velocity \( v_y \) at time \( t \) can be calculated using the equation:
\[ v_y = v_{y0} + at \]
Substituting the values:
\[ v_y = 0 + (1 \, \text{m/s}^2)(10 \, \text{s}) = 10 \, \text{m/s} \]

### Step 5: Calculate the Resultant Velocity
The resultant velocity \( v \) can be found using the Pythagorean theorem since the motion is along the x and y axes:
\[ v = \sqrt{v_x^2 + v_y^2} \]
Substituting the values:
\[ v = \sqrt{(10 \, \text{m/s})^2 + (10 \, \text{m/s})^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \, \text{m/s} \]

### Step 6: Calculate the Numerical Value
Calculating \( 10\sqrt{2} \):
\[ 10\sqrt{2} \approx 10 \times 1.414 = 14.14 \, \text{m/s} \]

### Final Answer
The speed of the body after \( 10 \) seconds is approximately \( 14.14 \, \text{m/s} \).

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