Magnetic Effects Of Current And Magnetism

Step 1: Understanding the Concept:
The force experienced by a charged particle moving in a magnetic field is given by the Lorentz force. The direction of this force can be determined using Fleming's Left-Hand Rule or the vector cross product.
Step 2: Key Formula or Approach:
The magnetic force on a charge moving with velocity in a magnetic field is given by:
The direction of the force is given by the direction of the cross product . We can use the right-hand rule for the cross product (since the charge is positive).
Step 3: Detailed Explanation:
Let's define the directions using a standard coordinate system:
- East is along the positive x-axis ( ).
- North is along the positive y-axis ( ).
- Vertically upwards is along the positive z-axis ( ).
From the problem statement:
- The velocity vector is horizontally eastwards: .
- The magnetic field vector is vertically upwards: .
Now, we calculate the cross product:
Using the cyclic property of unit vectors for cross product ( , , ), we know that .
The direction corresponds to the South.
Alternative Method (Fleming's Left-Hand Rule):
- Point your Forefinger in the direction of the Magnetic Field (Upwards).
- Point your Middle finger in the direction of the Current (or motion of positive charge, which is Eastwards).
- Your Thumb will point in the direction of the Force, which is towards the South.
Step 4: Final Answer:
The direction of the force on the positively charged particle is towards the South.

Step 1: Understanding the Concept:
The magnetic dipole moment ( ) of a current loop is a vector quantity that measures the strength and orientation of the magnetic field produced by the loop. Its magnitude is the product of the current flowing in the loop and the area enclosed by the loop.
Step 2: Key Formula or Approach:
The formula for the magnetic moment ( ) of a current loop is: where is the current and is the area of the loop.
For a circular loop of radius , the area is .
Therefore, the formula becomes: Step 3: Detailed Explanation:
Let the initial conditions be: Initial current =
Initial radius =
So, the initial magnetic moment is .
Now, the conditions are changed as follows: New current, (halved)
New radius, (doubled)
Let's calculate the new magnetic moment ( ) with these new values: Substitute the expressions for and : Rearrange the terms: Since , we can write: Step 4: Final Answer:
The new magnetic moment is twice the initial magnetic moment. Therefore, the magnetic moment of the loop will be doubled. Option (B) is correct.

Note: In a standard, non-relativistic cyclotron, the time period of revolution of a charged particle is independent of its speed and the radius of its orbit. This is the fundamental principle upon which the cyclotron operates. Therefore, none of the given proportionality options are correct. We will proceed by deriving the correct relationship.
Step 1: Understanding the Concept:
A cyclotron accelerates charged particles using a constant magnetic field and an oscillating electric field. The magnetic field forces the particle to move in a circular path, and the electric field accelerates it each time it crosses the gap between the two "dee" electrodes. The question asks for the relationship between the time period of one revolution (T) and the particle's speed (V).
Step 2: Key Formula or Approach:
The magnetic force ( ) on the charged particle provides the necessary centripetal force ( ) to keep it in a circular path.
The time period (T) is the time taken to complete one full circle, which is the circumference divided by the speed.
Step 3: Detailed Explanation:
From the force balance equation, we can find an expression for the radius :
Now, substitute this expression for the radius into the time period equation:
The speed cancels out from the numerator and the denominator:
This result shows that the time period depends only on the mass ( ) and charge ( ) of the particle, and the strength of the magnetic field ( ). It is independent of the particle's speed ( ) and the radius of its orbit ( ).
Step 4: Final Answer:
The time period is independent of the speed . The correct relationship is . Since this is not among the options, the question is flawed.

Step 1: Understanding the Concept:
A charged particle moving in a magnetic field experiences a magnetic force (Lorentz force) only if its velocity has a component perpendicular to the direction of the magnetic field. The magnitude and direction of this force determine the particle's trajectory.
Step 2: Key Formula or Approach:
The magnetic force on a particle with charge moving with velocity in a magnetic field is given by the vector cross product:
The magnitude of this force is given by:
where is the angle between the velocity vector and the magnetic field vector .
Step 3: Detailed Explanation:
In this specific case, the charged particle is "moving along the magnetic field." This means its velocity vector is parallel to the magnetic field vector .
When two vectors are parallel, the angle between them is 0 .
Let's calculate the magnitude of the force using this angle:
Since the value of is 0:
If the particle were moving anti-parallel to the field, the angle would be 180 , and is also 0, resulting in zero force.
Since the net magnetic force acting on the particle is zero, there is no change in its state of motion according to Newton's first law. The particle will continue to move with its constant velocity, which means it will follow a straight-line path.
Step 4: Final Answer:
Because the magnetic force on the particle is zero when it moves parallel to the magnetic field, its path will be an undeflected straight line.