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• Magnetic fields can arise from various sources
• Bar magnets
• Earth magnets
• Current-carrying systems
• Moving charges
• Varying electric fields

Bar Magnets

• Every magnet is essentially a magnetic dipole
• Pole strength represents the magnetic strength of a magnetic pole
• Pole strength is a scalar parameter
• Its unit is Amp x meter
• Imaginary magnetic field (B actual) ≈ 0.91
• Magnetic fields exist in closed loops only, while electric fields exist in open loops
• Single poles do not exist in isolation
• Formulas exist for calculating forces (F), electric fields (E), and electric force (Fe) for single poles, involving pole strengths (m1, m2), charges (q1, q2), distances (r), and constants (k, µ0)
• Magnetic moment (M) of a magnetic dipole or bar magnet is defined as M = m x l
• m Represents the pole strength and l is the effective length between magnetic dipoles (S-N)
• Magnetic moment is a vector quantity with the unit A x m^2
• The Bohr magneton (µB) has a value of approximately 0.923 x 10^-23 A x m^2

Magnetic Moment of a Magnet (M)

• If a magnet is cut into 'n' equal parts parallel to the magnetic axis
• The magnetic moment of each part is M' = M/n
• If a magnet is cut into 'n' equal parts perpendicular to the magnetic axis
• The magnetic moment of each part is M' = M/n
• When a thin magnet is bent
• The magnetic moment changes (M' ≠ M)
• When bending is performed, M' < M

Magnetic Fields Due to Dipoles (B)

• The magnetic field due to a dipole depends on the magnetic moment (m), distance (r), and position relative to the dipole

Axial/End-on/Longitudinal/Tan A Position

• Baxis = (µ0 / 4π) * (2M / r^3)
• The direction is along the magnetic moment vector (M)

Equator/Broadside/Transverse/Tan B Position

• Beq = (µ0 / 4π) * (M / r^3)
• The direction is opposite to the magnetic moment vector (M)

General Point (r, θ)

• B = (µ0 / 4π) * (M / r^3) * √(1 + 3cos²θ)
• The angle between B and M is θ + tan⁻¹(tanθ / 2)
• The magnetic potential (Vm) due to a dipole is Vm = (µ0 / 4π) * (Mcosθ / r²)

Behavior in an External Magnetic Field

• Net force (Fnet) is zero
• Torque (τ) = MBsinθ = M x B
• Potential energy (U) = -MBcosθ
• Work (W) = MB(cosθ₁ - cosθ₂)
• For small angular displacements, a dipole performs Simple Harmonic Motion (SHM) with a time period T = 2π√(I / MB), where I is the moment of inertia

Magnetic Needle

• A magnetic needle is a small bar magnet used to detect the direction of a magnetic field

Electric Fields vs Magnetic Fields

• The surface integral of the electric field is equal to electric charge density /permittivity of free space (Gauss's law in electrostatics)
• The surface integral of the magnetic field is zero, showing Gauss's theorem is not applicable/useful in magnetism
• It is not applicable because single poles do not exist
• Force (F) between two dipoles varies inversely with distance (r), and magnetic field due to dipole is proportional to distance (r)
• The magnetic effect of current was discovered, where magnetic fields are produced around current-carrying systems

Magnetic Field/Magnetic Flux Density/Magnetic Induction (B)

• Represents the strength of magnetism at a particular point and is a vector quantity
• Units for magnetic field
• Tesla (T)
• Weber per square meter (Wb/m²)
• Newton per Ampere per meter (N/A·m) CGS unit is Gauss (G)
• 1 T = 10^4 Gauss

Biot-Savart Law (BSL)

• Defines the magnetic field at a point due to a current element
• Magnetic field dB = (µ₀ / 4π) * (Idlsinθ / r²)
• µ₀ is permeability of free space, I is current, dl is length element, θ is the angle between dl and the position vector r
• Vector Form
• dB = (µ₀ / 4π) * (I dl x r̂ / r²)
• dB, dI and position vector are mutually perpendicular
• For straight current
• If thumb points along the current & fingers stretched towards the observation point
• B is perpendicular (I) to the right-hand palm
• For circular current
• Fingers folded along the circular current direction, the thumb indicates direction of B

Straight Current-Carrying Wire

Infinite Wire

• B = (µ₀I / 4πd) * (sinθ₁ + sinθ₂) angle depends on length

Semi-Infinite Wire

• Observation point in front of the fixed end
• B = µ₀I / 4πd

Magnetic Field Due to a Current-Carrying Coil

Center

• For circular current , Bcentre = (µ₀I / 2R)

Case A - Angle over radius

• B = (µ₀I / 4πR ) * Φ

Case B - Angle over 2π

• B= (µ₀I / 2πR) * θ

Magnetic Field at Axial Point of Current-Carrying Circular Ring

• On ring N, I, R
• B = (µ₀NIR² / 2(R² + x²)^(3/2)) = Bcentre / (1 + x²/R²)^(3/2))
• Bcentre means center of the ring
• At center (x = 0)
• B = µ₀NI / 2R
• Nearby point (x <<< R)
• B ≈ Bcentre * (1 - (3x² / 2R²))
• Faraway point (x >>> R)
• B = (µ₀NIR² / 2x³)

Circular Coil

• For two circular coils at center If the radius(R)same , current and terms are the same :
• B α N/R
• B1/B2 = N1/N2
• Circular coil made of fix length wire, If N2 turns and current Remains same:
• (M1/NI) = (M2/N2)^2
• For Helmeltz Pair, this system is used to produced Approx uniform magnetic field

Straight Current Carrying Infinite Wires

• Bnet = (µ₀d / 2)

Magnetic Field Intensity (H)

• Ratio of magnetic flux density & Magnetic permeability
• H = Bmed/µ = Bvacuum/µ₀
• Has some units ex:) Amp/meter, oersted

Ampere's Circuital Law (ACL)

• The line integral of B.dl over a closed loop is equal to µ₀ times the current enclosed by the loop
• The line integral
• ∫B⋅dl = µ₀ΣI
• Another form with Magnetomotive force (MMF)
• ∫H⋅dl = ΣI
• Third form with work done
• W / m = µ₀ΣI

Application of ACL

Infinite Wire/Long Wire

• B = (µ₀I / 2πr)

Solid Cylindrical Wire

Outside (r > R)

Boutside = (µ₀I / 2πr)

Inside (r < R)

• Binside = (µ₀Ir / 2πR²)

AT Axis (R = 0)

• Baxis =0

Solid / Hollow Cylindrical Pipe

• Formulas exist to compute the magnetic field outside/inside the wire

Long Cylindrical Current Carrying Ring Pipe

• Magnetic field at 'r' distance depends on the region and the inner/outer radius of the wiring
• At symmetrical closed loop made of uniform wire
• Geometrical center (B=0) in respective of length of parallel path

Circular Path of Uniform Wire

• the circuit as shown is current
• Bcentre is 0

Solenoid/Electromagnet

• The coil with some length has:
• Conducting wire is wound on cylindrical Insulating frame
• All turns are electrically Insulated With each other when current passed
• Almoset uniform Magnetic field is produced along, axis inside the frame
• System is equivalent to a group of Helmontz Pair
• Inside != 0
• Outside = 0 (ideally)

Solenoid Types

• Infinite/Long (R<<L)
• Finite (R<< L)

Finite Solenoid

• B = (µ₀ni / 2) * (cosα - cosβ)
• n is Turn density (turn per unit length)
• For infinite solenoid
• α = 0
• β= 180° ⇒ Baxis = µ₀nI

Toroid/Endless Solenoid

• The location means get जहाँ जाती है 'B' बरए है(Tire में)

Force Acting on Moving Charge in an External Magnetic Field

• A force results from magnetic field and change in magnetic state
• F = q(v x B)
  • q is changing parameter, v is velocity, B is magnetic field

Vector

• In the vector →'q' will be used along with the sign:
• Right-hand palm Rule is only * Applicable when →If v I 4
• Palm → direction of +q

Factors

• Force on a charge is always to its velocity
  • Only changes its direction not the Magnitude
• Work done is always zero, because it can't change Kinetics Energy
• Teacher is like Magnetic field can change the change direction only
• You text book influence the magnetic field

Motion of Charge

• Case where 'q' is directed on, the direction is straight

Time period

• Time period of Revolution and Tis Independent of "v":.

Angle

• "P" always equal to exit angle
• Horizontal"B" constant.
• If there in a electric field with B both helix increase:
• *A proton equation 11: radius increases

1. same speed li) same radius = radius
2. proton speed is accerate

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