A particle of mass m$\mathrm{m}$ and charge q$\mathrm{q}$ is fastened to one end A$\mathrm{A}$ of a massless string having equilibrium length $\ell$ , whose other end is fixed at point O$\mathrm{O}$ . The whole system is placed on a frictionless horizontal plane and is initially at rest. If uniform electric field is switched on along the direction as shown in figure, then the speed of the particle when it crosses the x-axis is

Core Logic

For the boundary limits matching a successful collision with the plate under maximum field conditions, the electron must be moving directly away from sheet$

Core Logic

For the boundary limits matching a successful collision with the plate under maximum field conditions, the electron must be moving directly away from sheet $Sinitially ($ initially ($u = +1 \mathrm{~m/s}). The accelerating field acts as a decelerating force, turning it around to hit the sheet exactly at$). The accelerating field acts as a decelerating force, turning it around to hit the sheet exactly at $t = 1 \mathrm{~s}.

Therefore, relative to the initial position vector pointing outwards:

• Initial speed$.

  Therefore, relative to the initial position vector pointing outwards:

  • Initial speed $u = 1 \mathrm{~m/s}
  • Net displacement at strike$
  • Net displacement at strike $S = -1 \mathrm{~m}(since it returns to the plate at origin)
  • Total time elapsed$ (since it returns to the plate at origin)
  • Total time elapsed $t = 1 \mathrm{~s}

  Step 1: Apply Second Equation of Motion

  Using$

Step 1: Apply Second Equation of Motion

Using $S = u t + \frac{1}{2} a t^2:

Step 2: Solve for Surface Charge Density

Equating this deceleration value to the electrostatic tracking acceleration:

Step 3: Extract alpha

Comparing this with the algebraic pattern$

Step 3: Extract alpha

Comparing this with the algebraic pattern $\alpha \left[\frac{m \varepsilon_0}{e}\right]:

$:

$\alpha = 8

Therefore, the value of$

Therefore, the value of $\alphais$ is $8.

Pattern Recognition

Be careful with coordinate signs. If the electron were moving towards the plate initially ($.

Pattern Recognition

Be careful with coordinate signs. If the electron were moving towards the plate initially ($u = -1), any field value would pull it in even faster, meaning the time would be less than$), any field value would pull it in even faster, meaning the time would be less than $1 \mathrm{~s}$. The 'maximum field' boundary constraint implies the electron is thrown away and turned around at its apex peak, mapping perfectly to a return displacement.

Chapter Mix

Class 12 Physics: Electric Charges and Fields Class 11 Physics: Motion in a Straight Line