A proton of charge +e and mass m enters a uniform magnetic Field `vecB=hatB` with an intial velocity `vecv=v_(0x)hati+v_(0y)hatj`. Find an expression in unit-vector notation for its velocity at any later time t.

Answer

To find the expression for the velocity of a proton moving in a uniform magnetic field at any later time \( t \), we start with the given information:

1. **Charge of the proton**: \( +e \)
2. **Mass of the proton**: \( m \)
3. **Initial velocity**: \( \vec{v} = v_{0x} \hat{i} + v_{0y} \hat{j} \)
4. **Magnetic field**: \( \vec{B} = B \hat{i} \)

### Step 1: Analyze the motion in the magnetic field

The magnetic force acting on a charged particle moving in a magnetic field is given by the Lorentz force equation:

\[ \vec{F} = q \vec{v} \times \vec{B} \]

In our case, the magnetic field \( \vec{B} \) is in the \( \hat{i} \) direction, and the velocity \( \vec{v} \) has components in both the \( \hat{i} \) and \( \hat{j} \) directions.

### Step 2: Calculate the magnetic force

Substituting the values into the Lorentz force equation:

\[ \vec{F} = e (v_{0x} \hat{i} + v_{0y} \hat{j}) \times (B \hat{i}) \]

Using the cross product properties:

\[ \hat{i} \times \hat{i} = 0 \quad \text{and} \quad \hat{j} \times \hat{i} = -\hat{k} \]

Thus, we have:

\[ \vec{F} = e v_{0y} \hat{j} \times B \hat{i} = e B v_{0y} (-\hat{k}) = -e B v_{0y} \hat{k} \]

### Step 3: Determine the acceleration

Using Newton's second law, \( \vec{F} = m \vec{a} \), we can find the acceleration:

\[ m \vec{a} = -e B v_{0y} \hat{k} \]

This gives:

\[ \vec{a} = -\frac{e B v_{0y}}{m} \hat{k} \]

### Step 4: Integrate to find velocity

The acceleration is constant in the \( \hat{k} \) direction, so we can integrate to find the velocity in the \( \hat{k} \) direction:

\[ v_z(t) = v_{0z} + a_z t = 0 - \frac{e B v_{0y}}{m} t = -\frac{e B v_{0y}}{m} t \]

### Step 5: Combine the components of velocity

The \( x \) and \( y \) components of the velocity remain unchanged because the magnetic field does not exert a force in those directions. Therefore, the velocity at time \( t \) can be expressed as:

\[ \vec{v}(t) = v_{0x} \hat{i} + v_{0y} \hat{j} + v_z(t) \hat{k} \]

Substituting \( v_z(t) \):

\[ \vec{v}(t) = v_{0x} \hat{i} + v_{0y} \hat{j} - \frac{e B v_{0y}}{m} t \hat{k} \]

### Final Expression

Thus, the expression for the velocity of the proton at any later time \( t \) is:

\[ \vec{v}(t) = v_{0x} \hat{i} + v_{0y} \hat{j} - \frac{e B v_{0y}}{m} t \hat{k} \]