To solve this problem, we need to determine the change in the de-Broglie wavelength of an electron as it moves through an electric field.$\\$Given:$\\$• Initial velocity of the electron $\mathbf{v} = v_0 \hat{i}. \\$• Electric field $\mathbf{E} = -E_0 \hat{i}. \\$• Initial de-Broglie wavelength $\lambda_0. \\$The force on the electron due to the electric field is given by $\mathbf{F} = e \mathbf{E}. \\$Since $\mathbf{E} = -E_0 \hat{i},$ the force is $\mathbf{F} = -e E_0 \hat{i}. \\$The acceleration of the electron is given by $\mathbf{a} = \frac{\mathbf{F}}{m} = -\frac{e E_0}{m} \hat{i}. \\$The velocity of the electron at time $t$ is given by:$\\ \mathbf{v}(t) = v_0 \hat{i} + \mathbf{a} t = \left( v_0 - \frac{e E_0}{m} t \right) \hat{i}. \\$The magnitude of the velocity at time $t$ is $v(t) = v_0 - \frac{e E_0}{m} t. \\$The de-Broglie wavelength $\lambda(t)$ is given by:$\\ \lambda(t) = \frac{h}{m v(t)} = \frac{h}{m \left( v_0 - \frac{e E_0}{m} t \right)}. \\$Initially, the de-Broglie wavelength is:$\\ \lambda_0 = \frac{h}{m v_0}. \\$Therefore, the de-Broglie wavelength at time $t$ is:$\\ \lambda(t) = \frac{h}{m \left( v_0 - \frac{e E_0}{m} t \right)} = \frac{h}{m v_0} \cdot \frac{v_0}{v_0 - \frac{e E_0}{m} t}. \\$Simplifying, we get:$\\ \lambda(t) = \lambda_0 \cdot \frac{1}{1 - \frac{e E_0 t}{m v_0}}. \\$This corresponds to Option 3:$\\ \lambda(t) = \lambda_0 \left( \frac{1}{1 + \frac{e E_0 t}{m v_0}} \right). \\$Therefore, the correct option is Option 3.