A proton moves with a velocity of \(5 \times 10^6 \hat{\mathbf{j} \mathrm{ms}^{-1}}\) through the uniform electric field, \(\mathbf{E}=4 \times 10^6[2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+0.1 \hat{\mathbf{k}}] \mathrm{Vm}^{-1}\) and the uniform magnetic field \(\mathbf{B}=0.2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+\hat{\mathbf{k}}] \mathrm{T}\). The approximate net force acting on the proton is

Given, speed of proton, \(v=5 \times 10^6 \hat{\mathbf{j}} \mathrm{m} / \mathrm{s}\) Electric field, \(\mathbf{E}=4 \times 10^6[2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+0.1 \hat{\mathbf{k}}] \mathrm{V} / \mathrm{m}\) Magnetic field, \(\mathbf{B}=0.2[\hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+\hat{\mathbf{k}}] \mathrm{T}\) Charge on proton, \(q=1.6 \times 10^{-19} \mathrm{C}\)
Net force acting on the proton is calculated according to Lorentz's force as
\(\mathbf{F}=q[\mathbf{E}+\mathbf{v} \times \mathbf{B}] \)
\(=1.6 \times 10^{-19}\left[4 \times 10^6(2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+0.1 \hat{\mathbf{k}})\right. \) \(\left.+5 \times 10^6 \hat{\mathbf{j}} \times 0.2(\hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+\hat{\mathbf{k}})\right] \)
\(=1.6 \times 10^{-19} \times 10^6[4(2 \hat{\mathbf{i}}+0.2 \hat{\mathbf{j}}+0.1 \hat{\mathbf{k}}) \) \( +(-\hat{\mathbf{k}}+0+5 \hat{\mathbf{i}})] \)
\(\Rightarrow \mathbf{F}=1.6 \times 10^{-13}[13 \hat{\mathbf{i}}+0.8 \hat{\mathbf{j}}-0.6 \hat{\mathbf{k}}] \)
\(\therefore \mathbf{F}=|\mathrm{F}|=1.6 \times 10^{-13} \sqrt{(13)^2+(0.8)^2+(0.6)^2} \)
\(=1.6 \times 10^{-13} \sqrt{170}=20.86 \times 10^{-13} \mathrm{~N} \)
\(=20 \times 10^{-13} \mathrm{~N}\)