Electromagnetic Induction and Magnetic Forces

Magnetism and Magnetic Fields

• Magnetism is the force of attraction or repulsion between objects caused by the motion of electric charges.

Magnetism

• Magnets attract when opposite poles (N and S) are next to each other.
• Magnets repel when like poles (N and N, or S and S) are next to each other.
• Either pole of a bar magnet attracts an unmagnetized object containing iron, such as a nail.
• A bar magnet sets up a magnetic field in the space around it, affecting other bodies.

Magnetic Fields

• A compass needle aligns with the magnetic field at its position.
• Earth possesses a magnetic field caused by currents in its molten core that changes over time
• Geological evidence indicates the Earth's magnetic field reverses direction at irregular intervals between 10,000 and 1,000,000 years.

Earth's Magnetic Field

• The geomagnetic north pole is actually a magnetic south (S) pole which attracts the north pole of a compass.
• Magnetic field lines indicate compass direction at a given location.
• Earth's magnetic field resembles that of a simple bar magnet, although it is caused by electric currents in the core.
• The Earth's magnetic axis is offset from its geographic axis.
• Breaking a bar magnet results in two magnets, each with a north and south pole.
• Magnetic poles always come in pairs and cannot be isolated, unlike electric charges.

Relationship of Magnetism to Moving Charges

• Discovered in 1820 by Hans Christian Oersted
• A compass needle is deflected by a current-carrying wire.
• Moving a magnet near a conducting loop induces a current in the loop, discovered by Michael Faraday and Joseph Henry.

Electric vs Magnetic Fields

• Electric fields are created by both stationary and moving charges
• Magnetic fields are created only by moving charges.
• Moving charges or currents create magnetic fields in the surrounding space
• Magnetic fields exert a force F on other moving charges or currents within the field.

Magnetic Force on Moving Charges

• Magnetic field (B) is a vector quantity associated with each point in space
• The direction of B at any position is the direction in which the north pole of a compass needle tends to point.
• Key Characteristics Include:
  • Magnitude is proportional to the magnitude of the charge.
  • Magnitude proportional to the "strength" of the field.
  • Depends on the particle's velocity (a charged particle at rest experiences no magnetic force).
  • The magnetic force F is always perpendicular to both B and the velocity v.
• The magnitude F of the force is proportional to the component of v perpendicular to the field
• When the component is zero (v and B are parallel or antiparallel), the force is zero.
• F = qvB sin φ, where q is the magnitude of the charge and φ is the angle between vectors v to B.
• The magnetic force F acting on a positive charge q moving with velocity v is perpendicular to both v and the magnetic field B.
• For given values of the speed v and magnetic field strength B, the force is greatest when v and B are perpendicular.
• The right-hand rule determines the direction of magnetic force on a positive charge moving in a magnetic field.

Determining direction of the force

• Place the v and B vectors tail to tail.
• Imagine turning v toward B in the v-B plane through the smaller angle.
• The force acts along a line perpendicular to the v-B plane
• Curl the fingers of your right hand around this line in the same direction as the rotation
• Your thumb now points in the direction the force acts.
• For negative charges, the force direction is opposite to that given by the right-hand rule.
• Two charges of the same magnitude but opposite sign moving with the same velocity in the same magnetic field experience magnetic forces equal in magnitude but opposite in direction.
• F = q(E + v × B) describes the total force F as the vector sum of electric and magnetic forces acting on a charged particle.
• The SI unit of B is the Tesla (T), equivalent to 1 N s / C m , or 1 N / A m.
• The Gauss (G) is another unit of B (1 G = 10-4 T).
• Earth's magnetic field is of the order of 10-4 T or 1 G.

Magnetic Field Lines and Flux

• Magnetic field lines represent magnetic fields.
• They start from the north pole and end on the south pole of a magnet.
• The line through any point is tangent to the magnetic field vector B at that point.
• Densely packed field lines indicate a stronger magnetic field.

Magnetic Flux

• Magnetic flux is quantified as the number of magnetic field lines passing through a surface
• Defined as dΦB = B⊥dA.
• ΦB = ∫B⊥dA = ∫B cos φ dA calculates the total magnetic flux through a surface
• B is uniform and the surface is a plane then ΦB = BA cos φ.
• The SI unit of magnetic flux is the weber (Wb), where 1 Wb = 1 T m2.
• Gauss's law for magnetism states the total magnetic flux through any closed surface is zero.

Motion of Charged Particles in Magnetic Fields

• A charged particle at point O, possessing charge q moving with velocity v in a uniform magnetic field B experiences simple motion
• The magnetic force F = qv × B has magnitude F = qvB and a certain direction
• The force is always perpendicular to v,
• The force cannot change the magnitude of the velocity, only its direction.
• Magnetic force never does work on the particle.
• A charged particle's motion is always motion with constant speed under the effect of a magnetic field

Circular Motion

• A charged particle travels in a circle of radius R, with centripetal acceleration v²/R.
• Equating magnetic force and Newton's second law = qvB = m(v²/R), solving for the radius gives:
  • R = mv / |q|B = p / |q|B describes the radius of a circular orbit where p is the magnitude of the particle's momentum.
• Angular speed ω of the particle in the circle where ω is v / R, can be rewritten as:
  • Equation: ω = |q|B / m.
• Frequency a particle is called the cyclotron frequency in this context.
• A uniform magnetic field does no work on the particle, and its speed and kinetic energy remain constant.
• Charged particles from the sun are trapped by Earth's magnetic field in doughnut-shaped regions (Van Allen radiation belts)

Applications of Motion of Charged Particles

• Velocity Selector - Select particles of same speed from a beam
• Accelerating charged particle with mass m, charge q, and speed v where electric and magnetic fields are perpendicular selects particles with zero total force.
• The speed for no deflection is when -qE + qvB = 0, is calculated as v = E/B from the formula above.

Thomson's experiment

• Performed to calculate the e/m of an electron
• Involves a highly evacuated glass container
• Electrons are emitted from the accelerating potential V between two anodes and two anodes (A and A')
• Resulting acceleration is the speed v of the electrons.
• This experiment can be satisfied combined with another equation to get SO(e / m) = E2 / 2VB2.

Magnetic Force on a Current-Carrying Conductor

• A straight conductor segment of length l and cross-sectional area A experiences a magnetic force in a uniform magnetic field B.
• The magnetic force formula is determined from several equations and laws:
  • F = (nAl)(qvB) = (nqvdA)(lB).
• A way to determine the current density is the equation J , the equation can further simplify into
  • F = IlB
  • F = Il × B quantifies the magnetic force on a straight wire segment.

Test your knowledge of electromagnetic induction with questions about induced emf, Lenz's Law, and eddy currents. Explore the effects of time-varying magnetic fields on conductors. Investigate magnetic forces between current-carrying wires and their interactions.