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 solve the problem of a proton entering a uniform magnetic field, we can follow these steps:

### Step 1: Understand the Given Information
We have a proton with:
- Charge \( e \)
- Mass \( m \)
- Initial velocity \( \mathbf{v} = v_x \hat{i} + v_y \hat{j} \)
- A uniform magnetic field \( \mathbf{B} = B \hat{i} \)

### Step 2: Determine the Force Acting on the Proton
The magnetic force acting on a charged particle moving in a magnetic field is given by:
\[ \mathbf{F} = q \mathbf{v} \times \mathbf{B} \]
For our case:
\[ \mathbf{F} = e (v_x \hat{i} + v_y \hat{j}) \times (B \hat{i}) \]
Since the cross product of parallel vectors is zero:
\[ \mathbf{F} = e (v_x \hat{i} + v_y \hat{j}) \times (B \hat{i}) = 0 + e v_y B \hat{k} = e v_y B \hat{k} \]
This force acts in the negative \( z \)-direction.

### Step 3: Analyze the Motion
- The \( x \)-component of the velocity \( v_x \) remains unchanged because there is no force acting in the \( x \)-direction.
- The \( y \)-component of the velocity will change due to the magnetic force acting on it.

### Step 4: Find the Change in Velocity
The acceleration \( \mathbf{a} \) in the \( y \)-direction is given by:
\[ \mathbf{a} = \frac{\mathbf{F}}{m} = \frac{e v_y B}{m} \hat{k} \]
This indicates that the proton will undergo circular motion in the \( yz \)-plane due to the force acting in the \( z \)-direction.

### Step 5: Determine the Angular Frequency
The angular frequency \( \omega \) of the circular motion is given by:
\[ \omega = \frac{qB}{m} = \frac{eB}{m} \]

### Step 6: Express the Motion in Terms of Time
The angle \( \theta \) at time \( t \) is given by:
\[ \theta = \omega t = \frac{eB}{m} t \]

### Step 7: Find the Components of Velocity at Time \( t \)
- The \( x \)-component remains constant:
\[ v_x(t) = v_x \]
- The \( y \)-component varies with time as:
\[ v_y(t) = v_y \cos\left(\frac{eB}{m} t\right) \]
- The \( z \)-component varies as:
\[ v_z(t) = -v_y \sin\left(\frac{eB}{m} t\right) \]

### Step 8: Combine the Components
The velocity vector at time \( t \) can be expressed as:
\[ \mathbf{v}(t) = v_x \hat{i} + v_y \cos\left(\frac{eB}{m} t\right) \hat{j} - v_y \sin\left(\frac{eB}{m} t\right) \hat{k} \]

### Step 9: Final Expression
Thus, the expression for the velocity of the proton at time \( t \) in unit vector notation is:
\[ \mathbf{v}(t) = v_x \hat{i} + v_y \cos\left(\frac{eB}{m} t\right) \hat{j} - v_y \sin\left(\frac{eB}{m} t\right) \hat{k} \]

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