A particle of mass `m = 0.1 kg` and having positive charge `q = 75 mu C` is suspended from a point by a thread of length `l = 10 cm`. In the space, a uniform horizontal electric field `E = 10^4 NC^-1` exists. The particle is drawn aside so that the thread becomes vertical and then it is projected horizontally with velocity v such that the particle starts to move along a circle with the same constant speed V. Calculate the radius of the circle and speed `v(g = 10 ms^-2)`.

Explanation

To solve the problem, we need to find the radius of the circular path and the speed of the charged particle moving in a uniform electric field. Here's a step-by-step solution:

### Step 1: Understanding the Forces Acting on the Particle
The particle of mass \( m = 0.1 \, \text{kg} \) and charge \( q = 75 \, \mu C = 75 \times 10^{-6} \, C \) is suspended by a thread of length \( l = 10 \, \text{cm} = 0.1 \, \text{m} \). The uniform electric field \( E = 10^4 \, \text{N/C} \) acts horizontally.

When the particle is displaced and then projected horizontally, it will move in a circular path at an angle \( \theta \) with respect to the vertical.

### Step 2: Determine the Angle of Equilibrium
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