Oersted's Experiment and Magnetic Fields

Oersted's Experiment

• A current-carrying conductor produces a magnetic field in its surrounding area.
• Direction of the magnetic field can be found using Ampere's swimming rule.
• The SI unit for magnetic field is Wm⁻² or Tesla (T).
• One tesla (T) equals 1 weber per square meter (Wbm⁻²) or 1 newton per ampere per meter (NA⁻¹m⁻¹).
• Gauss and oersted are CGS units of magnetic field; 1 gauss equals 10⁻⁴ tesla.

Maxwell's Cork Screw Rule

• The direction of the magnetic line of force is given by thumb's direction when a right-handed corkscrew is rotated, with the screw's tip pointing in the direction of the current.
• A dot represents a magnetic field coming out of the plane, while ® denotes a magnetic field perpendicular to the plane in the downward direction.

Ampere's Swimming Rule

• If a man is swimming along the wire in the direction of the current with his face towards the needle, the north pole of the magnetic needle deflects towards his left hand.

Magnetic Field

• It is the space around a magnet or current-carrying conductor where its magnetic influence is felt.

Biot Savart's Law

• The magnetic field dB produced by a current element Idl at a distance r is given by dB = (μ₀ / 4π) * (Idl * r / r³).
• Another expression is dB = (μ₀ / 4π) * (Idl sin θ / r²), where θ is the angle between current direction and free space's absolute permeability μ₀.
• The direction of magnetic field dB is the same as that of I dl * r.
• Magnetic field's SI unit is Wm⁻² (tesla), with the CGS unit being gauss or oersted, and 1 gauss = 10⁻⁴ tesla.

Magnetic Field Due to a Straight Current Carrying Conductor

• Magnetic field (B) is given by B = (μ₀ I / 4π r) * (sin φ1 + sin φ2), where φ1 and φ2 are angles subtended by the ends of the conductor at the observation point relative to the perpendicular.
• For an infinitely long conductor with observation point near the center, B = μ₀ / 4π * 2I / r.
• For an infinitely long conductor with observation point near one end, B = μ₀ / 4π * I / r.

Magnetic Field Lines

• They represent the magnetic B field in a region.
• They form closed, continuous curves that do not interact.
• Tangent at any point indicates the direction of the magnetic field.
• Outside a magnet, field lines go from north to south pole; inside, they go from south to north.
• Concentric circles centered on the conductor in a plane perpendicular to it represent the magnetic field lines around a straight current-carrying conductor
• The Right Hand Thumb Rule indicates the direction of magnetic field lines.
• Curling fingers indicate the direction of magnetic field lines if the thumb points in the direction of the current and holding a current-carrying conductor in the right hand.

Magnetic Field on the Axis of a Current Carrying Circular Coil

• The magnetic field at a distance x from the center on the axis of a circular coil: B = (μ₀ n I r²)/ (2(r² + x²)³/²), where r is the coil radius, n is the number of turns, and I is the current.
• At the coil's center, B = (μ₀ n I) / (2r).
• The face of the coil is south if current flows clockwise and north if current flows counter-clockwise.

Magnetic Dipole

• Each current carrying loop is a magnetic dipole with south (S) and north (N) poles, similar to a bar magnet
• Each magnetic dipole possesses a magnetic moment (M), with a magnitude of |M| = NiA, where N is the number of turns, i is the current, and A is the cross-sectional area of the loop.

Ampere's Circuital Law

• The line integral of magnetic field induction B around any closed path in vacuum equals μ₀ times the total current threading the closed path: ∮B ⋅ dl = μ₀I, μ₀ is the free space permeability and I is the current.
• The law applies to closed paths of any size/shape around a current-carrying conductor, independent of distance from the conductor.

Solenoid

• A solenoid is a tightly wound helix of insulated copper wire.
• Magnetic field inside a long solenoid: B = μ₀ n I (n is turns per unit length, I is current).
• Magnetic field at one end of a long solenoid: B = μ₀ n I / 2.

Toroid

• A toroid consists of numerous turns of copper wire wrapped around an anchor ring, forming a closed, endless solenoid in the shape of a ring.
• Magnetic field inside the toroid's turns: B = μ₀ n I.
• The magnetic field inside a toroid remains constant and is tangential to the circular closed path.
• Net current enclosed by the toroidal space is zero leading to a magnetic field of zero anywhere in its space.

Magnetic Field Due to a Current Carrying Long Circular Cylinder

• Outside the cylinder (r > R): B =μ / 2=l/r or B = µ/2πl/r
• Inside a thin metal sheet cylinder: B = 0.
• Inside the cylinder when current is uniformly distributed (r < R): B = μ₀ μr / (2 π Ir / R²), where μ₀ and μᵣ are permeabilities of free space and cylinder material.

Force Acting on a Charge Particle Moving in a Uniform Magnetic Field

• A charged particle moving in a magnetic field experiences a force F = q(v × B).
• The magnitude of this force is |F| = Bqv sin θ
• B represents magnetic field intensity
• q is the charge on the particle
• v denotes the particle's speed
• θ represents the angle between the magnetic field and motion direction.

Fleming’s Left Hand Rule

• If the thumb, forefinger, and center finger of the left hand are stretched mutually perpendicular, with the forefinger pointing in the direction of magnetic field, central finger in the direction of current, then the thumb points in the direction of magnetic force.

Lorentz Force

• Sum total force exerted on a charge as it travels inside electronic/magnetic field environments expressed as F = q(E * v * B)
• When a charged particle enters a magnetic field normally, it follows a circular path
• The path radius is r = mv / Bq, making r proportional to mv and inversely proportional to q/m
• Time period is T = 2πm / Bq
• A charged particle takes on a helical route when it enters a magnetic field an any angle except the 90°
• The route radius becomes r = mv sin θ / Bq
• The pitch: [ Pitch = T * v cos θ = 2πmv cos θ / Bq ] is defined as the length traveled by a charged particle over a period of time.

Cyclotron

• A cyclotron accelerates positively charged particles, using an electric field and a strong magnetic field.
• A positively charged particle gets accelerated through a modera electric field to cross it again and again by use of strong magnetic field.
• The radius of circular path is r = mv / Bq
• Cyclotron frequency is v = Bq / 2πm
• m and q stand for the mass and charge of the positive ion, while B refers to magnetic strength
• Maximum kinetic energy gained Emax = B²q²r₀² / 2m
• r₀ is the circular path's maximum radius.
• As the speed of the positive ion approaches the speed of light, its mass increases where = mass of the lOD
• maximum mass of the ion
• v = speed of Ion and
• c = speed of light.

Limitations of the Cyclotron

• Cyclotrons cannot accelerate uncharged particles like neutrons.
• Positively charged particles with large masses cannot move without speed limitations within a cyclotron.

Force Between Two Infinitely Long Parallel Current Carrying Conductors

• The force is F= Ho 121 / 2π Γ

Torque on a Current Carrying Coil in a Uniform Magnetic Field

• Torque experienced by a current carrying coil in uniform magnetic field: τ = NBIA sin θ
• N is the number of coil turns, B the magnetic field strength, I the current, A the area of coil cross section.
• 0 angle between the magnetic field and the plane of the coil.

Moving Coil Galvanometer

• This tool detects and measures current levels
• Deflecting torque is the same as restoring torque during system equilibrium.

Current Sensitivity

• A galvanometer's current sensitivity: Is = θ / I = NBA / K
• θ angle of twist,
• I current,
• N number of coil turns,
• B magnetic field strength
• A area of the coils cross section

Voltage Sensitivity

• It is the deflection produced for each unit voltage across galvanometer ends expressed as Vs = θ / V = NBA / KR.
• R represents galvanometer coil resistance

A Sensitive Galvanometer has...

• A large number of turns (N)
• High magnetic field (B)
• Large cross-sectional area (A)
• Small restoring torque per unit twist (K)

Ammeter

• Ammeters have relatively low resistance and measure electric flow inside circuit environments
• These always connect in series mode
• Low resistance runs in parallel to galvanometers when its converted to ammeter.

Galvanometer Conversion to an Ammeter

• To convert a galvanometer (resistance G) to an ammeter, connect a low resistance S in parallel.
• Required low resistance S in parallel form : S = Ig / I-Ig G

Ideal Ammeter

• The resistance is zero

Voltmeter

• High-resistance galvanometers used for measuring potential difference connecting to two spots.
• It is connected in parallel.

Ideal Voltmeter

• The resistance is infinity

Galvanometer Conversion to Voltmeter

• To convert a galvanometer to a voltmeter, connect a high resistance in its respective series.
• High resistance R is given by the formula R= V / Ig - G, where V = Ig (G + R).

Explore Oersted's experiment demonstrating magnetic fields around current-carrying conductors. Learn to determine field direction using Ampere's swimming rule, and understand Tesla, Gauss, and Oersted units. Covers Maxwell's corkscrew rule too.