Magnetic Fields



  • Out of the screen/paper: out notation
  • Into the screen/paper:in notation

Magnetic field created by a current-carrying wire

Consider the following current-carrying wire: current carrying wire

  • General equation describing this system:

\left| \overrightarrow { B } \right|= \frac{\mu I}{2 \pi r}

  •  Variables:
    • \overrightarrow{B} = magnetic field created by moving charge
      • \left| \overrightarrow { B } \right| = magnitude of the magnetic field at a point in space outside the wire (SI unit: \mathrm{T = N} / \mathrm{Am})
    • \mu = permeability the material of the wire is made of (SI unit: \mathrm{H} / \mathrm{m})
    • I = electric current (SI unit: \mathrm{A = C} / \mathrm{s})
    • r = distance from the current-carrying wire (SI unit: \mathrm{m})

Magnetic Force: two currents in the same direction

Consider two identical current-carrying wires with charge flowing through them in the same direction: two currents same direction

  • Variables
    • R = distance between the two wires
    • \overrightarrow{B}_i and \left\| \overrightarrow{L}_i \right\| denote the magnetic field and length of wire i (respectively)
      • Importantly, for the purposes of this article, the direction \overrightarrow{L}_i is defined as the direction current I_i is traveling
      • Note: we treat I_i as a scalar value
  • Let \overrightarrow{F}_{12} denote the magnetic force of I_1 on I_2
    • Then, \overrightarrow{F}_{12} = I_2 \overrightarrow{L}_2 \times \overrightarrow{B}_1
    • Using right-hand rule convention,  this implies that the direction of \overrightarrow{F}_{12} is pointed purely in the horizontal direction towards the right
  • Similarly, if we let \overrightarrow{F}_{21} deonote the magnetic force of I_2 on I_1 then…
    • \overrightarrow{F}_{21} = I_1 \overrightarrow{L}_1 \times \overrightarrow{B}_2
    • Using the right-hand rule convention, this implies that the direction of \overrightarrow{F}_{21} is pointed purely in the horizontal direction towards the left 3
  • Summary of this example:
    • \overrightarrow{B}_1 is exerting a downward force on I_2
    • \overrightarrow{B}_2 is exerting an upward force on I_1
    • \overrightarrow{F}_{12} and \overrightarrow{F}_{21} directions indicate that both wires will become attracted towards eachother
      • This will cause R (i.e., the distance between the wires) to decrease
      • As R decreases , the wires wil accelerate towards eachother

Induced current in a wire

Consider the following magnetic field \overrightarrow{B} (purple) popping out of the page and a wire laid over the field so that it is overlapping \overrightarrow{B} for a distance L induced current in a wire - Copy

  • Let Q (orange) be a charge positioned at the lower end of the wire (SI units: Coulombs)

Nothing happens when Q is stationary

  • This is because the force \overrightarrow{F} due to the magnetic field is equal to the cross product of Q times the velocity v of the charge and \overrightarrow{B}:

\overrightarrow{F} = Q \overrightarrow{v} \times \overrightarrow{B}

  • Recall: the magnitude of the cross product can be written as…

\left| \overrightarrow{F} \right| = Q \left| \overrightarrow{v} \right| \cdot \overrightarrow{B} \sin {\theta}

In our diagram, the wire and the magnetic field are perpendicular to eachother

  • Recall: \sin {90^\circ} = 1
  • Thus, for this instance,

\left| \overrightarrow{F} \right| = Q \left| \overrightarrow{v} \right| \cdot \overrightarrow{B}

  • For our stationary charge, \left| \overrightarrow{v} \right| = 0
    • Thus, \left| \overrightarrow{F} \right| = 0

The work done by a magnetic field per charge in a coductive wire can be described by the following equation:

\frac{W}{Q} = L(\overrightarrow{v} \times \overrightarrow{B})

  • Importantly, the output for this equation has units in Joules/Coulomb
    • Recall: 1 \mathrm{Volt} = 1 J / C
    • **Thus, the work done by the magnetic field on a charge causes the charge to move AS IF there is a potential differe4nce of 1 \mathrm{V} over the length of the wire that overlaps with \overrightarrow{B}
      • In this context, we aren’t saying the charge is moving due to a difference in potential energy (like it does in a simple DC circuit)
      • Even though differences in potential energy give rise to voltage, in teh context of our example, it is the magnetic field giving rise to the voltage
        • Specifically, when a voltage is induced from a magnetic field, it is said that the magnetic field is exerting an electromotive force or emf
    • emf has units in J/C
      • For circuit analysis, this has the same effect as a potential (or voltage) difference
      • We can define the emf mathematically as \mathrm{emf} = L \overrightarrow{v} \times \overrightarrow{B}
        • Recall: we can pull constants out of a cross product to rewrite this equation as \mathrm{emf} = L(\overrightarrow{v} \times \overrightarrow{B})


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