An electron has velocity `v = (2.0 xx 10^6 m/s) hati + (3.0 x 10^6 m/s )hatj`. Magnetic field present in the region is `B = (0.030 T) hati - (0.15 T) hatj`.
(a) Find the force on electron.
(b) Repeat your calculation for a proton having the same velocity.

Explanation

To solve the problem, we will break it down into two parts: (a) finding the force on the electron and (b) finding the force on the proton.

### Part (a): Force on the Electron

1. **Identify the Given Values**:
- Velocity of the electron:
\[ \mathbf{v} = (2.0 \times 10^6 \, \text{m/s}) \hat{i} + (3.0 \times 10^6 \, \text{m/s}) \hat{j} \]
- Magnetic field:
\[ \mathbf{B} = (0.030 \, \text{T}) \hat{i} - (0.15 \, \text{T}) \hat{j} \]
- Charge of the electron:
\[ q = -1.6 \times 10^{-19} \, \text{C} \]

2. **Calculate the Cross Product \(\mathbf{v} \times \mathbf{B}\)**:
- Set up the determinant for the cross product:
\[ \mathbf{v} \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2.0 \times 10^6 & 3.0 \times 10^6 & 0 \\ 0.030 & -0.15 & 0 \end{vmatrix} \]

- Calculate the determinant:
\[ \mathbf{v} \times \mathbf{B} = \hat{i} \left(3.0 \times 10^6 \cdot 0 - 0 \cdot (-0.15)\right) - \hat{j} \left(2.0 \times 10^6 \cdot 0 - 0 \cdot 0.030\right) + \hat{k} \left(2.0 \times 10^6 \cdot (-0.15) - 3.0 \times 10^6 \cdot 0.030\right) \]
\[ = 0 \hat{i} - 0 \hat{j} + \hat{k} \left(-0.30 \times 10^6 - 0.09 \times 10^6\right) \]
\[ = \hat{k} \left(-0.39 \times 10^6\right) \]

3. **Calculate the Force on the Electron**:
- The magnetic force \(\mathbf{F}\) on the electron is given by:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
- Substituting the values:
\[ \mathbf{F} = -1.6 \times 10^{-19} \cdot (-0.39 \times 10^6) \hat{k} \]
\[ = 6.24 \times 10^{-14} \hat{k} \, \text{N} \]

### Part (b): Force on the Proton

1. **Identify the Given Values**:
- The velocity of the proton is the same as that of the electron.
- Charge of the proton:
\[ q = +1.6 \times 10^{-19} \, \text{C} \]

2. **Calculate the Force on the Proton**:
- Using the same cross product calculated earlier:
\[ \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) \]
- Substituting the values:
\[ \mathbf{F} = 1.6 \times 10^{-19} \cdot (-0.39 \times 10^6) \hat{k} \]
\[ = -6.24 \times 10^{-14} \hat{k} \, \text{N} \]

### Final Answers:
- (a) The force on the electron is:
\[ \mathbf{F}_{\text{electron}} = 6.24 \times 10^{-14} \hat{k} \, \text{N} \]
- (b) The force on the proton is:
\[ \mathbf{F}_{\text{proton}} = -6.24 \times 10^{-14} \hat{k} \, \text{N} \]