Magnetic Field

Answer (d)

When a charged particle enters a uniform magnetic field, the direction of the magnetic force is perpendicular to the velocity of the particle. The magnetic force causes the particle to move in a circular or helical path if the velocity has a component perpendicular to the field. However, if the initial velocity is parallel to the magnetic field (i.e., along the direction of the magnetic field), the magnetic force will be zero because the angle between the velocity and the magnetic field is , and thus: Since no magnetic force acts on the particle, it will continue to move in a straight line along the direction of the magnetic field.

Step 1: Understanding the Concept:
Torque on a current loop: , where is angle between and normal to plane.

Step 2: Detailed Explanation:
If is in the plane of the loop, normal is perpendicular to , so , .


Step 3: Final Answer:
Torque = .

Step 1: Understanding the Concept:
Biot-Savart law: .

Step 2: Detailed Explanation:
The source of magnetic field is the current element , which is a vector quantity.

Step 3: Final Answer:
Magnetic field is produced by vector source .

Concept:

Step 1: Observation.
Frequency independent of radius.

Step 2: Conclusion.
Changing frequency does not affect radius.

Concept: Magnetic force acts perpendicular to velocity.

Step 1: Velocity components.

• Parallel component unchanged
• Perpendicular component circular motion

Step 2: Resultant motion.
Combination helical path

Concept:

Step 1: Substitute values.


Step 2: Find angle.

Step 1: Understanding the Concept:
This is a standard formula for the magnetic field at the center of a circular current-carrying loop.

Step 2: Detailed Explanation:
The magnetic field at the center of a circular loop of radius carrying a current is given by: where is the permeability of free space. The direction is perpendicular to the plane of the loop (right-hand rule).

Step 3: Final Answer:
The magnetic field is .

Step 1: Understanding the Concept:
The magnetic field inside a long solenoid is uniform and given by , where is the number of turns per unit length.

Step 2: Detailed Explanation:
Given: Current, . Length of solenoid, . Total number of turns, . Number of turns per unit length, . Permeability of free space, . Magnetic field, . Approximately .

Step 3: Final Answer:
The magnetic field is .

Step 1: Understanding the Concept:
An ideal solenoid is a long coil of wire. When current flows through it, it generates a nearly uniform and strong magnetic field concentrated strictly along its interior central axis.

Step 2: Key Formula or Approach:
The magnitude of the magnetic field ( ) well inside a long solenoid is given by Ampere's Law:

Where:
= permeability of free space = T m/A
= number of turns per unit length (turns/meter)
= current in amperes

Step 3: Detailed Explanation:
Given values:
Turn density, turns/m = m
Current, A
Substitute these into the formula:

Notice how the terms cleanly cancel out:

Group the numbers and powers of 10:


To write this in standard scientific notation:


Step 4: Final Answer:
The magnetic field is T.

Step 1: Analyze the motion of a charged particle in a magnetic field.
The path of a charged particle in a uniform magnetic field is determined by the angle between its velocity vector and . The force is given by .

Step 2: Analyze the different cases based on the angle .
Case 1: or ( is parallel or anti-parallel to ). The cross product is zero, so the force . The particle moves in a straight line. Option (A) is incorrect because if is vertical (parallel to ), the path is a straight line, not a circle.
Case 2: ( is perpendicular to ). The force is constant in magnitude and is always perpendicular to . This provides the necessary centripetal force for uniform circular motion. Option (B) is incorrect because if is horizontal (perpendicular to the vertical ), the motion is circular, not helical.
Case 3: is neither parallel nor orthogonal to . The velocity vector can be resolved into two components: one parallel to ( ) and one perpendicular to ( ). The parallel component does not interact with the magnetic field (as ), so the particle moves with constant velocity along the field direction.
The perpendicular component causes a magnetic force that leads to circular motion in the plane perpendicular to . The combination of this linear motion along the field and circular motion perpendicular to it results in a helical (spiral) path.

Step 3: Evaluate the options.
Based on the analysis in
Step 2:
- (A) is incorrect.
- (B) is incorrect.
- (C) correctly describes the condition for helical motion.
- (D) and (E) are incorrect as the motion can be a straight line, circular, or helical depending on the initial velocity's direction.


Final Answer: (C) make helical motion if v is neither parallel nor orthogonal to B

Step 1: Recall the formula for magnetic force on a charged particle.
The magnetic force (Lorentz force) experienced by a charged particle moving in a magnetic field is given by the formula: Where: - is the magnetic force vector. - is the charge of the particle (for an electron, ). - is the velocity vector of the particle. - is the magnetic field vector.

Step 2: Apply the condition for a stationary electron.
The problem states that the electron is stationary. This means its initial velocity is zero.

Step 3: Calculate the magnetic force.
Substitute into the Lorentz force equation: Since the cross product of the zero vector with any other vector is the zero vector, the magnetic force on the stationary electron is zero.

Step 4: Conclusion.
According to Newton's first law, an object at rest stays at rest if no net force acts on it. Since the magnetic force on the stationary electron is zero, it will not accelerate and will continue to remain stationary.

Final Answer: (A) remains stationary

Step 1: Understanding the Concept:
In a cyclotron, charged particles are accelerated by an alternating electric field and kept in a spiral path by a constant magnetic field. The cyclotron frequency is the frequency of the alternating electric field, which must match the frequency of revolution of the particle to ensure continuous acceleration.

Step 2: Key Formula or Approach:
The cyclotron frequency ( ) depends on the particle's charge ( ), its mass ( ), and the applied magnetic field ( ):


Step 3: Detailed Explanation:
"Identical conditions" implies that the magnetic field is the same for both particles.
For a proton ( ):
Charge
Mass
Cyclotron frequency
For an alpha particle ( ), which is a helium nucleus (2 protons, 2 neutrons):
Charge
Mass (mass of 4 nucleons)
Cyclotron frequency
Now, find the ratio of their frequencies ( ):


The ratio of the cyclotron frequency of an particle to that of a proton is .

Step 4: Final Answer:
The ratio is .

Concept:
• Magnetic field at centre:

Step 1: Same wire length
Length constant

Step 2: Substitute in formula


Step 3: For
Final Conclusion:
Option (D)

Concept:
• Radius in magnetic field:

Step 1: Given
Momentum same

Step 2: Relation


Step 3: Ratio
Final Conclusion:
Option (C)