Moving Charges And Magnetism

Step 1: Ampere's Circuital Law.
Ampere's circuital law states that the line integral of the magnetic field around a closed loop is proportional to the total current passing through the loop. The law is given by: where: - is the permeability of free space, - is the enclosed current.
Step 2: Magnetic Field due to a Straight Conductor.
Consider a long straight conductor carrying a current . We choose a circular path of radius around the wire. Since the magnetic field due to a straight conductor is radially symmetric, the magnetic field at a distance from the wire is tangential to the circle. The line integral of the magnetic field around the circular path is: Solving for , we get the magnetic field at a distance from the conductor:

Step 1: Magnetic Field Due to a Current-Carrying Conductor.
The magnetic field at a point due to a current in a long straight conductor is given by Ampere's law: where:
- is the permeability of free space ( ),
- is the current,
- is the distance from the wire to the point where the field is being measured.
Step 2: Magnetic Field at Point P Due to Conductor Y.
The magnetic field at point P due to conductor Y, with current and distance , is:
Step 3: Magnetic Field at Point P Due to Conductor X.
For the field to be zero at point P, the magnetic field due to conductor X must be equal and opposite in direction to the field due to conductor Y. Let the current in conductor X be , and the distance from point P to conductor X is . The magnetic field at point P due to conductor X is:
Step 4: Setting the Fields Equal.
For the magnetic fields to cancel each other out, we set : Solving for :
Step 5: Conclusion.
Thus, the current in conductor X must be in the opposite direction to cancel the magnetic field at point P.

Step 1: Definition of 1 Ampere.
1 ampere is defined as the current that, when flowing through each of two parallel conductors placed 1 meter apart in a vacuum, produces a force of per meter of length between the conductors.
Step 2: Explanation.
This definition is based on the force between two current-carrying conductors. The force per unit length between two parallel conductors is given by: where: - is the force per unit length, - and are the currents in the two conductors, - is the distance between the conductors, - is the permeability of free space ( ). For two conductors with a current of 1 ampere each, placed 1 meter apart, the force between them is . This is the basis for the definition of 1 ampere.

Step 1: Formula for Period of Revolution.
The period of revolution of a charged particle moving in a magnetic field is given by the formula: where is the mass of the particle, is the charge of the particle, and is the magnetic field strength.
Step 2: Mass and Charge of the Proton and -Particle.
- For the proton, the charge (where is the elementary charge) and mass is the mass of a proton. - For the -particle, the charge (twice the charge of a proton) and mass (four times the mass of a proton).
Step 3: Ratio of Periods.
The ratio of the periods for the proton and for the -particle is given by:
Step 4: Conclusion.
The ratio of the period of revolution of the -particle to the proton is 2:1.

Step 1: Biot-Savart Law.
The Biot-Savart law gives the expression for the magnetic field produced by a small current element. It is given by: where: - is the infinitesimal magnetic field produced by the current element, - is the current through the conductor, - is the infinitesimal length vector of the current element, - is the unit vector from the current element to the point of observation, - is the distance from the current element to the point of observation, - is the permeability of free space.
Step 2: Magnetic Field Due to a Current Carrying Circular Loop.
The magnetic field at the center of a circular loop of radius , carrying a current , is given by the formula: This expression is derived using the Biot-Savart law by integrating over the entire loop.
Step 3: Magnetic Moment of the Current Loop.
The magnetic moment of a current loop is given by: where: - is the current, - is the area of the loop. For a circular loop, the area , so the magnetic moment becomes:

Step 1: Formula for the Radius of the Circular Path.
The radius of the circular path of a charged particle in a magnetic field is given by: where is the mass of the particle, is the velocity, is the charge, and is the magnetic field strength.
Step 2: Analyzing the Proportionality.
We can express momentum as , so the radius becomes: Therefore, the radius is directly proportional to the momentum of the particle.
Step 3: Conclusion.
Hence, the correct answer is (B) momentum of the particle.

Step 1: Gauss’ Theorem.
Gauss’ theorem states that the total electric flux through a closed surface is equal to the charge enclosed divided by the permittivity of free space :
Step 2: Electric Field Due to a Thin Charged Wire.
Consider an infinitely long, straight charged wire with linear charge density (charge per unit length). To find the electric field at a distance from the wire, we use a cylindrical Gaussian surface with radius and length . The electric flux through the surface is given by: Since the charge enclosed is , applying Gauss' law: Solving for :
Final Answer:
The intensity of the electric field at a distance from a thin charged wire of infinite length is:

Step 1: Definition of Magnetic Moment.
Magnetic moment ( ) is the measure of the strength and orientation of a magnetic source. For a moving charge, it is given by the formula: Where:
- is the current,
- is the area enclosed by the moving charge.
For an electron moving in a circular path, we can express the magnetic moment as: Where:
- is the mass of the electron ( ),
- is the velocity of the electron ( ),
- is the radius of the circular orbit ( ).
Step 2: Calculation of Magnetic Moment.
Substitute the values into the formula for magnetic moment:
Step 3: Unit of Magnetic Moment.
The SI unit of magnetic moment is (ampere square meter).
Final Answer:
The magnetic moment of the electron is .

Step 1: Use the Photoelectric Equation.
The energy of the incident photons can be related to the work function and stopping potential using the photoelectric equation: Where:
- is the energy of the photon,
- is the work function,
- is the stopping potential,
- is Planck's constant,
- is the speed of light,
- is the wavelength of light.
Step 2: Rearranging the Equation.
Substitute in the equation:
Step 3: Convert eV to Joules.
1 eV = , so:
Step 4: Solving for Wavelength .
Now, solve for : Substitute the values of and :
Final Answer:
The wavelength of light is .

Step 1: Biot-Savart Law.
The Biot-Savart law gives the magnetic field produced at a point by a small current element. The law is given by: where: - is the infinitesimal magnetic field, - is the permeability of free space, - is the current, - is the infinitesimal length element of the wire, - is the unit vector from the current element to the point of observation, - is the distance from the current element to the point of observation.
Step 2: Unit of .
Rearranging the Biot-Savart law to isolate : From this, we can see that the units of are derived as follows: Thus, the unit of is , where is the tesla, is meters, and is amperes.
Step 3: Conclusion.
The unit of is .

The kinetic energy of the electron is given as 45 eV. We can convert this into joules: The expression for the kinetic energy of an electron is also: Where: - is the mass of the electron,
- is the velocity of the electron.
Rearranging for , we get: Now, using the formula for the radius of the circular path in a magnetic field: Where: - is the radius,
- is the mass of the electron,
- is the velocity of the electron,
- is the charge of the electron,
- is the magnetic field strength.
Substituting the known values: - ,
- ,
We get: Thus, the radius of the circular path is .

Step 1: Fleming's Left Hand Rule.
Fleming's Left Hand Rule is used to find the direction of force exerted on a moving charged particle in a magnetic field. It states that if you stretch the thumb, index finger, and middle finger of your left hand at right angles to each other: - The index finger points in the direction of the magnetic field , - The middle finger points in the direction of the velocity of the particle, - The thumb will point in the direction of the force on the charged particle.
Step 2: Formula for magnetic force.
The force on a charged particle moving in a magnetic field is given by the Lorentz force law: where: - is the charge of the particle, - is the velocity of the particle, - is the magnetic field, - denotes the vector cross product.
Step 3: Cross product calculation.
We are given: To find the magnetic force, we calculate the cross product : Expanding this determinant: Thus:
Step 4: Force on the particle.
Now, we use the formula for the force:
Step 5: Conclusion.
The magnitude of the force is , and the direction is along the negative -axis (perpendicular to both and ).

Step 1: Biot-Savart Law.
The magnetic field at the center of a current-carrying circular coil is derived using the Biot-Savart law. The Biot-Savart law states that the magnetic field at a point due to an infinitesimal current element is: where: - is the permeability of free space, - is the current, - is the infinitesimal length element of the wire, - is the unit vector from the current element to the point where the magnetic field is being calculated, - is the distance from the current element to the point.
Step 2: Magnetic field at the center of the coil.
For a circular loop of radius , the magnetic field at the center of the loop due to a current is given by: where: - is the magnetic field at the center of the coil, - is the permeability of free space, - is the current flowing through the coil, - is the radius of the coil.
Step 3: Conclusion.
Thus, the intensity of the magnetic field at the center of a current-carrying circular coil is:
Step 4: Law used.
The law used to derive this expression is the Biot-Savart law.