Magnetic Force on a Moving Charge
- Source: A magnetic field exerts a force on a moving charge.
- Direction: The force is perpendicular to both the velocity of the charge and the magnetic field. This direction is determined by the right-hand rule.
- Magnitude: The magnitude of the force is proportional to the charge, the velocity, the magnetic field strength, and the sine of the angle between the velocity and the magnetic field.
- Formula: F = qvBsinθ where: * F is the magnetic force * q is the charge * v is the velocity of the charge * B is the magnetic field strength * θ is the angle between v and B
Motion of a Charged Particle in a Magnetic Field
- Uniform Magnetic Field: A charged particle moving perpendicular to a uniform magnetic field experiences a centripetal force, causing it to move in a circular path. The radius of the circular path is given by:
- r = mv/qB
- Helical Motion: If the charged particle has a velocity component parallel to the magnetic field, it will follow a helical path.
Magnetic Force on a Current-Carrying Conductor
- Force: A current-carrying conductor in a magnetic field experiences a force.
- Direction: The direction of the force is given by Fleming’s left-hand rule.
- Magnitude: The magnitude of the force is proportional to the current, the length of the conductor in the magnetic field, the magnetic field strength, and the sine of the angle between the conductor and the magnetic field.
- Formula: F = BILsinθ
Torque on a Current Loop in a Magnetic Field
- Torque: A current loop in a magnetic field experiences a torque.
- Orientation: The torque tends to align the plane of the loop perpendicular to the magnetic field.
- Magnitude: The magnitude of the torque is proportional to the current, the area of the loop, the magnetic field strength, and the sine of the angle between the normal to the loop and the magnetic field.
- Formula: τ = NIABsinθ where: * τ is the torque * N is the number of turns in the loop * I is the current * A is the area of the loop * B is the magnetic field strength * θ is the angle between the normal to the loop and B
Magnetic Dipole Moment
- Definition: The magnetic dipole moment (μ) of a current loop is a measure of its strength as a magnetic source.
- Formula: μ = NIA
- Torque: The torque on a magnetic dipole in a magnetic field is given by:
- τ = μBsinθ
Biot-Savart Law
The Biot-Savart law describes the magnetic field produced by a current element:
- Field: Each current element creates a magnetic field at a point.
- Direction: The direction of the magnetic field is given by the right-hand rule.
- Magnitude: The magnitude of the magnetic field is proportional to the current, the length of the current element, the sine of the angle between the current element and the line joining the element to the point, and inversely proportional to the square of the distance between the element and the point.
- Formula: dB = (μ₀/4π) (Idlsinθ)/r²
Ampere’s Circuital Law
Ampere’s circuital law relates the magnetic field around a closed loop to the current passing through the loop:
- Line Integral: The line integral of the magnetic field around a closed loop is proportional to the total current enclosed by the loop.
- Applications: Ampere’s law is useful for calculating magnetic fields in situations with high symmetry.
Applications of Moving Charges and Magnetism
The principles of moving charges and magnetism are applied in numerous devices and technologies, including:
- Electric motors: Convert electrical energy into mechanical energy.
- Generators: Convert mechanical energy into electrical energy.
- Transformers: Change the voltage of alternating current.
- Mass spectrometers: Separate ions based on their mass-to-charge ratio.
- Magnetic Resonance Imaging (MRI): Uses magnetic fields and radio waves to create detailed images of the human body.