First slide

Electric field

Question

A ball of mass 2 kg, charge $1 C$ is dropped from top of a high tower. In space electric field exist in horizontal direction away from tower which varies as $E = (5 - 2 x) {Vm}^{- 1}$ . Find maximum horizontal distance (in m) that the ball can go from the tower.

Moderate

Solution

$\begin{array}{l} a_{x} = \frac{q E}{m} \\ \Rightarrow \frac{d v_{x}}{d t} = \frac{q}{m} (5 - 2 x) \end{array}$
$\begin{aligned} \Rightarrow v \frac{dv}{dx} = \frac{q}{m} (5 - 2 x) \\ \Rightarrow \frac{v_{x}^{2}}{2} = \frac{q}{m} (5 x - x^{2}) \\ \Rightarrow v_{x}^{2} = \frac{2 q}{m} (5 x - x^{2}) \end{aligned}$
$\Rightarrow \frac{dx}{dt} = \sqrt{\frac{2 q}{m} (5 x - x^{2})}$
For x to be maximum $\frac{dx}{dt} = 0$
$\begin{array}{l} \Rightarrow 5 x - x^{2} = 0 \\ \Rightarrow x = 5 m \end{array}$

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