Motion In Combined Electric And Magnetic Fields

Magnetic Field Due to a Small Current Element Using the Biot-Savart Law

To find the magnetic field at a point due to a small current element, we use the Biot-Savart law:

Where:

• is the small element of the wire,
• is the position vector from the current element to the point where the field is being calculated,
• is the distance from the element to the point.

Step 1: Setup and Vector Analysis

Given:

• The current ,
• The position vector of the point ,
• The element of the wire is 1 cm or , aligned along the X-axis, so .

The distance is:

Step 2: Applying the Biot-Savart Law

Now, we calculate :

The cross product simplifies to:

Thus,

Step 3: Integration

To find the total magnetic field, we integrate over the length of the wire element, which extends from to :

Step 4: Solving the Integral

On solving, the result gives:

Thus, the magnetic field at the point is .

When a charged particle moves in a magnetic field at a perpendicular angle to the field, it follows a circular path. The radius of this circular path is given by the formula: Where:
- is the mass of the particle,
- is the speed of the particle,
- is the charge of the particle,
- is the magnetic field strength. We know the following:
- The particle gains a speed after being accelerated through a potential difference .
- The magnetic field .
- The energy gained by the particle is equal to the work done by the electric field, which can be expressed as: From this equation, we can solve for (mass of the particle) in terms of (charge of the particle) and : Substituting this expression for into the formula for : Now, substitute the given values: - , - , - : Thus, the radius of the circular path described by the particle is 5 cm. Therefore, the correct answer is option (B).