Question

In the product \(\vec{F}=q(\vec{v}\times\vec{b})\) 
=\(q\vec{v}\times(B\hat i+B \hat j+B_0\hat k)\)
 For q=1 and \(\vec v = 2\hat i+4\hat j+6\hat k\) and \(\vec F=4\hat i-20\hat j+12\hat k\)
What will be the complete expression for \(\vec B\)  ?

• A. \(6\hat i+6\hat j-8\hat k\)
• B. \(-8\hat i-8\hat j-6\hat k\)
• C. \(-6\hat i-6\hat j-8\hat k\)
• D. \(8\hat i+8\hat j-6\hat k\)

Answer 1

The Correct Option is C

To solve the given problem, we need to determine the complete expression for the magnetic field vector \(\vec{B}\). We are given the formula for the magnetic force:

\(\vec{F} = q (\vec{v} \times \vec{B})\)

Here, the following values are provided:

• \(q = 1\)
• \(\vec{v} = 2\hat{i} + 4\hat{j} + 6\hat{k}\)
• \(\vec{F} = 4\hat{i} - 20\hat{j} + 12\hat{k}\)

The expression for \(\vec{B}\) is:

\(\vec{B} = B_i\hat{i} + B_j\hat{j} + B_k\hat{k}\)

To find \(\vec{B}\), we need to solve the equation:

\(\vec{F} = \vec{v} \times \vec{B}\)

By setting up the cross product of \(\vec{v}\) and \(\vec{B}\), we get:

	\(\hat{i}\)	\(\hat{j}\)	\(\hat{k}\)
\(\vec{v} =\)	2	4	6
\(\vec{B} =\)	B_i	B_j	B_k

Using the determinant method for the cross product, we find:

\(\vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 6 \\ B_i & B_j & B_k \end{vmatrix} = \hat{i}(4B_k - 6B_j) - \hat{j}(2B_k - 6B_i) + \hat{k}(2B_j - 4B_i)\)

Equating this to \(\vec{F}\), we have:

\(4\hat{i} - 20\hat{j} + 12\hat{k} = (4B_k - 6B_j)\hat{i} - (2B_k - 6B_i)\hat{j} + (2B_j - 4B_i)\hat{k}\)

By comparing coefficients, we get the system of equations:

1. \(4B_k - 6B_j = 4\)
2. \(2B_k - 6B_i = 20\)
3. \(2B_j - 4B_i = 12\)

Solving these equations step-by-step:

• From the first equation: \(B_k = \frac{6B_j + 4}{4}\)
• From the second equation: \(B_k = \frac{20 + 6B_i}{2}\)
• From the third equation: \(B_j = \frac{4B_i + 12}{2}\)

Solving this system for \(B_i\), \(B_j\), and \(B_k\):

1. Using equation 3: \(B_j = 2B_i + 6\)
2. Substitute \(B_j\) in equation 1: \(4B_k - 6(2B_i + 6) = 4\)

Solving gives:

\(B_i = -6, B_j = -6, B_k = -8\)

Thus, the correct option is:

\(-6\hat{i} - 6\hat{j} - 8\hat{k}\)