Question

A charged particle of mass \(1 \mathrm{~kg}\) and charge \(2 \mu \mathrm{c}\) is thrown from a horizontal ground at an angle \(\theta=45^{\circ}\) with speed $20 \mathrm{~m} / \mathrm{s}\(. In space a horizontal electric field \)\mathrm{E}=2 \times 10^{7} \mathrm{~V} / \mathrm{m}$ exist. The range on horizontal ground of the projectile thrown is $\ldots \ldots \ldots$ (A) \(100 \mathrm{~m}\) (B) \(50 \mathrm{~m}\) (C) \(200 \mathrm{~m}\) (D) \(0 \mathrm{~m}\)

Short answer

The range of the projectile for the charged particle in the presence of the electric field is \(100 \mathrm{m}\).

Step-by-step solution

Find the Vertical and Horizontal Components of Initial Velocity

We have the initial speed \(v_{0} = 20 \mathrm{m/s}\) and the angle of projection \(\theta = 45^{\circ}\). The components of initial velocity in the horizontal and vertical directions are: \[v_{0x} = v_{0} \cos \theta \] \[v_{0y} = v_{0} \sin \theta \] Find the values of \(v_{0x}\) and \(v_{0y}\).

Calculate Time of Flight

The time of flight for a projectile is the time taken for it to reach the ground again. Since the vertical component of velocity and acceleration are only affected by gravity, we can calculate the time of flight using this equation: \[t_{f}=\frac{2v_{0y}}{g}\] where \(t_f\) is the time of flight and \(g\) is the acceleration due to gravity (\(g = 9.81 \mathrm{m/s^2}\)). Calculate the time of flight.

Find Vertical Force and Acceleration Due to Electric Field

The electric field influences the charged particle with a force in the vertical direction: \[F_{y} = Eq\] where \(E\) is the electric field, \(q\) is the charge of the particle, and \(F_{y}\) is the vertical force. In this case, \(E = 2 \times 10^{7} \mathrm{V/m}\) and \(q = 2 \times 10^{-6} \mathrm{C}\). Next, calculate the acceleration in the vertical direction due to this force: \[a_{y} = \frac{F_{y}}{m}\] where \(a_y\) is the vertical acceleration and \(m = 1\mathrm{kg}\) is the mass of the particle. Calculate the value of \(a_{y}\).

Adjust the Time of Flight in the Presence of Electric Field

Since there is an upward acceleration due to the electric field, the time of flight will be affected. The adjusted time of flight can be calculated by finding the effective gravitational acceleration in the presence of the electric field: \[g_{eff} = g - a_{y}\] Now compute the new time of flight with the effective gravitational acceleration: \[t_{f_{eff}} = \frac{2v_{0y}}{g_{eff}}\] Calculate the adjusted time of flight \(t_{f_{eff}}\).

Calculate Range of the Projectile

The range of the projectile is the horizontal distance traversed during the time of flight. In the presence of the electric field, it can be calculated using: \[R_{eff} = v_{0x} \cdot t_{f_{eff}}\] Determine the range \(R_{eff}\) and compare it to the given choices. The solution should provide the correct range for the charged particle in the presence of the electric field.