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Question:

In a certain region static electric and magnetic fields exist. The magnetic field is given by B⃗ =B₀(î + 2ĵ + ḵ). If a test charge has velocity v⃗ =v₀(3î + ĵ + 2ḵ) and experiences no force in that region, then the electric field in the region, in SI units, is :

E⃗ =−v₀B₀(14ĵ+7ḵ)

E⃗ =v₀B₀(14ĵ+7ḵ)

E⃗ =−v₀B₀(î+ĵ+ḵ)

E⃗ =−v₀B₀(3î−2ĵ+ḵ)

Solution:

B⃗ =B₀(î+2ĵ+ḵ)
v⃗ =v₀(3î−ĵ+2ḵ)
Now,
B⃗ ×v⃗ =B₀v₀(−ḵ−2ĵ+4î−2ĵ)=B₀v₀(4î−4ĵ−ḵ)=−B₀v₀(14ĵ+7ḵ)