A proton of charge `e` and mass `m` enters a uniform magnetic field `B = Bi` with an initial velocity `v=v_xhati+v_yhatj`. Find an expression in unit vector notation for its velocity at time `t`.

Explanation

To find the expression for the velocity of a proton in a uniform magnetic field at time \( t \), we can follow these steps:

### Step 1: Understand the problem
The proton has an initial velocity given by:
\[ \mathbf{v} = v_x \hat{i} + v_y \hat{j} \]
and it enters a uniform magnetic field:
\[ \mathbf{B} = B \hat{i} \]
We need to find the velocity of the proton at time \( t \).

### Step 2: Determine the force acting on the proton
The force \( \mathbf{F} \) on a charged particle moving in a magnetic field is given by the Lorentz force equation:
\[ \mathbf{F} = q \mathbf{v} \times \mathbf{B} \]
For a proton, the charge \( q = e \). Since the magnetic field \( \mathbf{B} \) is along the \( \hat{i} \) direction, we can calculate the cross product.

### Step 3: Calculate the cross product
Using the initial velocity:
\[ \mathbf{v} = v_x \hat{i} + v_y \hat{j} \]
The cross product \( \mathbf{v} \times \mathbf{B} \) becomes:
\[ \mathbf{F} = e (v_x \hat{i} + v_y \hat{j}) \times (B \hat{i}) \]
Since \( \hat{i} \times \hat{i} = 0 \), we only need to consider the \( \hat{j} \) component:
\[ \mathbf{F} = e v_y B (\hat{j} \times \hat{i}) = -e v_y B \hat{k} \]

### Step 4: Write the equations of motion
The force leads to an acceleration \( \mathbf{a} \):
\[ \mathbf{a} = \frac{\mathbf{F}}{m} = -\frac{e v_y B}{m} \hat{k} \]
This means that the \( z \)-component of the velocity will change due to this force, while the \( x \)-component remains constant.

### Step 5: Find the velocity components at time \( t \)
The \( x \)-component of velocity \( v_x \) remains unchanged:
\[ v_x(t) = v_x \]

For the \( y \)-component, since there is no force acting in the \( y \)-direction, it also remains unchanged:
\[ v_y(t) = v_y \]

For the \( z \)-component, we can integrate the acceleration to find the velocity:
\[ v_z(t) = v_z(0) - \frac{e B}{m} v_y t \]
Assuming the initial \( z \)-component of velocity \( v_z(0) = 0 \):
\[ v_z(t) = -\frac{e B}{m} v_y t \]

### Step 6: Combine the components into a vector
Now we can express the total velocity at time \( t \):
\[ \mathbf{v}(t) = v_x \hat{i} + v_y \hat{j} - \frac{e B}{m} v_y t \hat{k} \]

### Final Expression
Thus, the expression for the velocity of the proton at time \( t \) in unit vector notation is:
\[ \mathbf{v}(t) = v_x \hat{i} + v_y \hat{j} - \frac{e B}{m} v_y t \hat{k} \]

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