Physics

Magnetic Field

Magnetic field concept, Lorentz force law, Biot-Savart law, and practical applications.

The magnetic field is manifested in the region of space around a magnet, a moving electric charge, or an electric current.
It is a vector field of forces characterized by the magnetic field vector, magnetic induction, or simply magnetic field. It is represented by B⃗\vec{\textbf{B}}, and its SI unit is the tesla (T).
Magnetic Field Lines:
  • They emerge from the north pole and enter the south pole of a magnet.
  • They form closed loops; that is, in a magnet, they go from the north pole to the south pole on the outside, and from south to north on the inside.
Magnetic field lines
Geek3, CC BY-SA 3.0, via Wikimedia Commons
A moving charge within a magnetic field experiences a force.
F⃗=q(v⃗×B⃗)\vec{\textbf{F}} = q(\vec{\textbf{v}} \times \vec{\textbf{B}})
The expression includes a cross product, so to determine the direction of the force, we must use the right-hand rule. The magnitude of the force is given by:
F=q⋅v⋅B⋅sin⁡θF = q \cdot v \cdot B \cdot \sin\theta
Where θ\theta is the angle between the velocity vector and the magnetic field vector.
If an electric field is present in addition to the magnetic field, the total force on the charge will be:
F⃗=q(E⃗+v⃗×B⃗)\vec{\textbf{F}} = q(\vec{\textbf{E}} + \vec{\textbf{v}} \times \vec{\textbf{B}})
Right-hand rule for Lorentz force
A moving charge generates a magnetic field around it. According to the Biot-Savart Law:
B⃗=μ4πq(v⃗×u⃗)r2\vec{\textbf{B}} = \frac{\mu}{4\pi} \frac{q(\vec{\textbf{v}} \times \vec{\textbf{u}})}{r^2}
The magnitude of the magnetic field is:
B=μ⋅q⋅v⋅sin⁡θ4πr2B = \frac{\mu \cdot q \cdot v \cdot \sin\theta}{4\pi r^2}
The direction of the magnetic field is perpendicular to the plane formed by v⃗\vec{\textbf{v}} and u⃗\vec{\textbf{u}} (unit vector from q to P). Its direction is given by the right-hand rule (curl fingers from v⃗\vec{\textbf{v}} to u⃗\vec{\textbf{u}}).
μ\mu is a characteristic constant of each medium called magnetic permeability. In a vacuum, the permeability value is:
μ0=4π⋅10−7 T⋅mA\mu_0 = 4\pi \cdot 10^{-7} \, \frac{\text{T} \cdot \text{m}}{\text{A}}
Magnetic field generated by a moving charge
Motion of a charge in a magnetic field, with velocity perpendicular to the field.
It describes uniform circular motion: the central force is the Lorentz force, and the motion is perpendicular to the B\textbf{B} field, so the magnitude of the force is: F=q⋅v⋅BF = q \cdot v \cdot B.
This force is equal to the centripetal force;
q⋅v⋅B=mv2R  ⟹  R=m⋅vq⋅Bq \cdot v \cdot B = m \frac{v^2}{R} \quad \implies \quad R = \frac{m \cdot v}{q \cdot B}
Circular motion of a charge in a magnetic field
Velocity selector: for a charge to move in a straight line through a region with mutually perpendicular electric and magnetic fields, the net force acting on it is:
F⃗=m⋅a⃗=(E⋅q−q⋅v⋅B)j⃗\vec{\textbf{F}} = m \cdot \vec{\textbf{a}} = (E \cdot q - q \cdot v \cdot B)\vec{\textbf{j}}
It must be satisfied that: a⃗=0  ⟹  E⋅q=q⋅v⋅B  ⟹  v=EB\vec{\textbf{a}} = 0 \implies E \cdot q = q \cdot v \cdot B \implies v = \frac{E}{B}
Velocity selector
Mass spectrometer. Separates isotopes of an element. An ionized atom is accelerated by a potential difference ΔV\Delta V, acquiring a velocity equal to:
v=2⋅q⋅ΔVmv = \sqrt{\frac{2 \cdot q \cdot \Delta V}{m}}
The charged particle enters a magnetic field B and is deflected into a circular path of radius R, with different values for isotopes of different mass.
R=m⋅vq⋅Bwhere the mass-to-charge ratio ismq=R2⋅B22⋅ΔVR = \frac{m \cdot v}{q \cdot B} \quad \text{where the mass-to-charge ratio is} \quad \frac{m}{q} = \frac{R^2 \cdot B^2}{2 \cdot \Delta V}
Mass spectrometer
Magnetic force on a current-carrying conductor: Current is the charge flowing per unit time: I=q/tI=q/t. The force experienced by a section of a straight wire of length L is:
F⃗=L(I⃗×B⃗)with magnitude F=I⋅L⋅B⋅sin⁡θ\vec{\textbf{F}} = L(\vec{\textbf{I}} \times \vec{\textbf{B}}) \quad \text{with magnitude } F = I \cdot L \cdot B \cdot \sin\theta
Where θ\theta is the angle between I\textbf{I} and B\textbf{B}, and the direction is given by the right-hand rule.
Magnetic force on a wire
Magnetic field created by a long straight current-carrying wire: Direction according to the right-hand rule and magnitude:
B=μ⋅I2πRB = \frac{\mu \cdot I}{2\pi R}
Magnetic field of a straight wire
Jfmelero, CC BY-SA 4.0, via Wikimedia Commons
Force between two parallel currents: two parallel, infinite wires, separated by a distance dd, carrying currents I1I_1 and I2I_2.
F=μ⋅I1⋅I22π⋅dLF = \frac{\mu \cdot I_1 \cdot I_2}{2\pi \cdot d} L
Parallel currents in the same direction attract each other. If they are in opposite directions, they repel.
Force between parallel current-carrying wires
Magnetic field created by a circular current loop: the field at the center is:
B=μ⋅I2RB = \frac{\mu \cdot I}{2R}
The direction of the vector B is given by the right-hand rule (curl fingers in direction of current, thumb points to B).
Magnetic field of a current loop
Magnetic field created by a solenoid:
B=N⋅μ⋅IlB = \frac{N \cdot \mu \cdot I}{l}
Where N is the number of turns in the solenoid and l is its length.
Magnetic field of a solenoid
Goodphy, CC BY-SA 4.0, via Wikimedia Commons