Lecture 5: Magnetic Fields and Moving Charges

Magnetic Fields and Moving Charges

Magnetic Materials & Facts

• Certain materials are magnetic:
  • Iron, cobalt, nickel, gadolinium are prominent.
  • Alloys can be used with some trickery.
• Magnets:
  • Can repel or attract without touching.
  • Opposites attract, likes repel.
  • Tiny magnets (compasses) align with the magnetic field.

Magnetism and Navigation

• Magnets were used as navigational tools.
  • Allowed navigation without landmarks or stars.
  • Important for ocean navigation.
• Magnets point in a north-seeking direction.
• Earth as a magnet:
  • Implies a powerful magnet within the Earth.
  • The geographic North Pole is the magnetic South Pole, and vice versa.
  • Earth has a magnetic field surrounding it.

Magnetic Fields (B)

• Earth is surrounded by a magnetic field.
• Magnetic field denoted by $B$, a vector with direction.
• Visualization using iron filings:
  • Filings align with the magnetic field lines.
  • Field goes from North to South.
• Field patterns:
  • Horseshoe magnet: field goes from north to south around the magnet.
  • Similar shape to the electric field between positive and negative charges.
• Relationship between electric and magnetic fields:
  • No effect observed when bringing a strong magnet near a highly charged object.
  • Unless the charge is moving.

Force on Moving Charges

• Experiment: electron beam (green line) being bent by magnets.
• Force occurs when velocity and field are perpendicular.
• Intrinsically a three-dimensional concept.
• If velocity and field are in the same direction, no force.
• Right-hand rule to determine the direction of force.
• The force on a moving charge in a magnetic field is given by: $F = q (v \times B)$.
  • $F$ is the force vector.
  • $q$ is the magnitude of the charge.
  • $v$ is the velocity vector of the charge.
  • $B$ is the magnetic field vector.
  • $\times$ denotes the cross product.
• Direction:
  • Point fingers in the direction of velocity ($v$).
  • Curl fingers from $v$ into the field $B$.
  • Thumb points in the direction of the force ($F$).

Units

• B (magnetic field) measured in Tesla (T).
• If $q$ is in Coulombs (C) and $v$ is in meters per second (m/s), then $F$ is in Newtons (N).
• Teslas are large units.
• MRI machines have fields of a few Tesla.

Conventions for Representing 3D Fields

• Fields into the page: represented by X (tail of an arrow).
• Fields out of the page: represented by . (tip of an arrow).
• Example: v going up, B going into the page, F goes to the left.

Concept Question 1

• Positive charge comes out of the screen; magnetic field points up from the keyboard.
• Force is to the left.

Force on Current-Carrying Wires

• Moving charges are commonly seen in currents in wires.
• Force equation is $F = qvB$.
• If the velocity is uniform, $v = \frac{l}{t}$, where $l$ is length and $t$ is time.
• Rewrite equation as $F = \frac{q}{t}lB$.
• Since current $I = \frac{q}{t}$, then $F = IlB$.
• A length $l$ of wire in a field $B$ will experience a force if it's carrying a current $I$.

Demonstration: Force on Current-Carrying Wire

• Ingredients:
  • Magnet (magnetic field).
  • Current-carrying wire (length $l$ between magnet poles).
  • Battery, switch.
• Passing current through the wire while it is between the poles of the magnet causes the wire to jump.

Magnetic Fields Created by Wires

• Magnets interact with each other through their magnetic fields.
• A wire must have a magnetic field to interact with a magnet.
• Sprinkling iron filings around a wire shows circular field patterns.

Right-Hand Rule #2

• Thumb in direction of current.
• Fingers show the direction of the magnetic field lines.
• Magnitude:
  • $B$ is proportional to $I$ (current).
  • $B$ is inversely proportional to $r$ (distance from wire).
• $B = \frac{\mu_0 I}{2 \pi r}$
  • $\mu_0 = 4 \pi \times 10^{-7} \text{ Tesla meters per Ampere}$

Concept Question 2

• Long wire bent back upon itself like a hairpin.
• Segments repel each other.
• Top wire creates a magnetic field (right-hand rule).
• B going into the screen.
• v is to the left.
• Force is downward.
• Another demo showing wires repelling and attracting based on current direction.

Concept Question 3

• Hairpin wire: current doubles.
• Force goes up four times.
• Field doubles from top wire.
• Lower wire has twice the current.
• Two factors of two give an overall factor of four.

Solenoids

• Looping a wire around creates a larger magnetic field in the center.
• Solenoid: coils of wire.
• When current flows through it:
  • Current generates a magnetic field.
  • Fields align up.
  • It becomes a magnet on demand (when switch is on).

Right-Hand Rule #3

• Fingers in the direction of the current.
• Thumb in the direction of the North Pole.

Practical Example: Battery-Operated Doorbell

• Close the switch.
• Battery activates.
• Current flows through the electromagnet.
• Magnetic field pulls on a piece of metal.
• The piece of metal moves, releasing the contact area.
• Inertia causes the hammer to strike the bell.
• Hammer springs back, makes contact, repeats.

Ampere's Law

• Analog to Gauss's law.
• Gauss's law discovers charge; Ampere's law discovers current.
• With Gauss's law, we had an $\epsilon<em>0$. With Ampere's law, we have a $\mu</em>0$.
• Add up field parallel to a line segment, not a surface.
• Look at an enclosed line, not a surface.
• You pick what closed circle: rectangle, square, or circle
• It is a two dimensional figure, not a three dimensional figure.

Natural Magnetism

• Electrons have spin (angular momentum about an axis).
• Spinning charge (electron) generates a magnetic field.
• Electrons pair up with opposite spins, canceling out.
• Unpaired spins in certain elements (iron, cobalt) give rise to natural magnetism.
• These atoms act as little teeny atomic magnets.

Ferromagnetism

• Atoms of iron, cobalt act as tiny magnets.
• If not mutually aligned, ferromagnetic material, not a permanent magnet.
• They can get aligned in molten iron by bringing up an external magnet, as it starts to cool the iron atoms in the melt begin to be biased into a common direction.
• Magnetic materials become magnetic by aligning regions known as magnetic domains.
• Bulk material: unaligned individual regions, unmagnetized.
• As they get aligned you begin to make a magnet.
• Domains: regions composed of billions of atoms.
• Alignment of random regions makes things magnetic.
• Disalignment loses you out on magnetism.
• Electron Microscopy can visualize the regions.

Magnetic Domains & Noise

• Super sensitive microphone next to a piece of material.
• As domains align, they make little vibrational noises (microphone can pick up).