A particle with charge 2.0 C movess through a uniform magnetic field. At one instant the velocity of the particle is `(2.0hati+4.0hatj+6.0hatk)` m`//`s and the magnetic force on the particle is `(4.0hati-20hatj+12hatk)` N.
The x and y components of the magnetic fields are equal. What is `vecB` ?

Answer

To find the magnetic field vector \(\vec{B}\) given the information in the problem, we will follow these steps:

### Step 1: Define the magnetic field vector
Since the x and y components of the magnetic field are equal, we can express the magnetic field vector as:
\[ \vec{B} = x \hat{i} + x \hat{j} + z \hat{k} \]

### Step 2: Write the force equation
The magnetic force \(\vec{F}\) on a charged particle moving in a magnetic field is given by the equation:
\[ \vec{F} = q (\vec{v} \times \vec{B}) \]
where \(q\) is the charge of the particle, \(\vec{v}\) is its velocity, and \(\vec{B}\) is the magnetic field.

### Step 3: Substitute known values
Given:
- Charge \(q = 2.0 \, \text{C}\)
- Velocity \(\vec{v} = (2.0 \hat{i} + 4.0 \hat{j} + 6.0 \hat{k}) \, \text{m/s}\)
- Force \(\vec{F} = (4.0 \hat{i} - 20.0 \hat{j} + 12.0 \hat{k}) \, \text{N}\)

We can substitute these values into the force equation:
\[ (4.0 \hat{i} - 20.0 \hat{j} + 12.0 \hat{k}) = 2.0 \, (\vec{v} \times \vec{B}) \]

### Step 4: Calculate the cross product \(\vec{v} \times \vec{B}\)
Using the determinant method for the cross product:
\[ \vec{v} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 4 & 6 \\ x & x & z \end{vmatrix} \]

Calculating the determinant, we have:
\[ \vec{v} \times \vec{B} = \hat{i} (4z - 6x) - \hat{j} (2z - 6x) + \hat{k} (2x - 4x) \]
\[ = (4z - 6x) \hat{i} - (2z - 6x) \hat{j} - 2x \hat{k} \]

### Step 5: Substitute back into the force equation
Now substituting \(\vec{v} \times \vec{B}\) back into the force equation:
\[ (4.0 \hat{i} - 20.0 \hat{j} + 12.0 \hat{k}) = 2.0 \, ((4z - 6x) \hat{i} - (2z - 6x) \hat{j} - 2x \hat{k}) \]
This simplifies to:
\[ (4.0 \hat{i} - 20.0 \hat{j} + 12.0 \hat{k}) = (8z - 12x) \hat{i} - (4z - 12x) \hat{j} - 4x \hat{k} \]

### Step 6: Equate coefficients
Now we equate the coefficients of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\):
1. For \(\hat{i}\):
\[ 8z - 12x = 4 \quad \text{(1)} \]
2. For \(\hat{j}\):
\[ -4z + 12x = -20 \quad \text{(2)} \]
3. For \(\hat{k}\):
\[ -4x = 12 \quad \text{(3)} \]

### Step 7: Solve the equations
From equation (3):
\[ x = -3 \]

Substituting \(x = -3\) into equation (1):
\[ 8z - 12(-3) = 4 \implies 8z + 36 = 4 \implies 8z = 4 - 36 \implies 8z = -32 \implies z = -4 \]

### Step 8: Write the magnetic field vector
Now substituting \(x\) and \(z\) back into the magnetic field vector:
\[ \vec{B} = -3 \hat{i} - 3 \hat{j} - 4 \hat{k} \]

### Final Answer
Thus, the magnetic field vector is:
\[ \vec{B} = -3.0 \hat{i} - 3.0 \hat{j} - 4.0 \hat{k} \, \text{T} \]