Magnetic Fields 9702

A Level Physics 9702, 20 Magnetic Fields

what’s a magnetic field

• field of force

• produced either by

  • moving charges

  • permanent magnets

what materials produces magnetic fields

• permanent magnets produce magnetic fields

• a stationary charge will not produce a magnetic fields

represent a magnetic field by field lines

• used to represent direction and magnitude

• always directed from north pole to south pole

• magnetic field lines are closer together at the poles where the field is strongest

• magnetic field lines never cross

magnetic field between two bar magnets

• two like poles repel each other

• two opposite poles attract each other

uniform magnetic field

• magnetic field strength is same at all points

• represented by equally spaced parallel lines

magnetic field & wire

• magnetic field is created around a current carrying wire due to the flow of electrons

force on a current-carrying conductor

• force might act on a current-carrying conductor placed in a magnetic field

• the current-carrying conductor produces its own magnetic field

  • when interacting with an external magnetic field, it’ll produce a force

formula for magnetic force on a current-carrying conductor

• The force F on a conductor carrying current I at an angle θ to a magnetic field with flux density B is defined by the equation

F = BIL sin θ

• Where:

  • F = force on a current-carrying conductor in a B field (N)

  • B = magnetic flux density of applied B field (T)

  • I = current in the conductor (A)

  • L = length of the conductor (m)

  • θ = angle between the conductor and applied B field (degrees)

force on conductor can be increased by

• increasing

  • strength of magnetic field

  • current flowing through conductor

  • length of conductor in the field

magnetic flux density, B

• The force exerted per unit current per unit length on a straight current-carrying conductor placed perpendicular to the magnetic field

• measured in Teslas (T)

Tesla (T)

• the flux density that causes a force of 1N on a 1m wire carrying a current of 1A at right angles to the magnetic field

higher magnetic flux density means…

• higher flux density, stronger magnetic field - region where field lines are closer together

• lower flux density, weaker magnetic field - region where field lines are further apart

when will a current-carrying conductor experience max. magnetic force?

• if current through it is perpendicular to direction of field lines

  • sinθ = 1

when will a current-carrying conductor experience no magnetic force?

• it will experience no force if the current in the conductor is parallel to the field

Fleming’s left hand rule

• the magnetic force acting on conductor, the B-field, and current are all perpendicular to each other

  • thumb, direction of motion of force F on conductor

  • first finger, direction of applied magnetic field B

  • second finger, direction of flow of conventional current (from +ve to -ve)

representing magnetic fields in 3D

• represents the magnetic field

  • out of the page - dots

  • into page - crosses

magnetic force on a moving charge

• a moving charge produces its own magnetic field

  • when interacting with an applied magnetic field, it will experience a force

formula for magnetic force, F, of an isolated particle

• F = BQvsinθ

• F = magnetic force on isolated particle (N)

• Q = charge of the particle (C)

• B = magnetic flux density (T)

• v = speed of the particle (m s^-1)

• θ = angle between charge’s velocity and magnetic field (degrees)

direction on force on a moving charge

• depends on

  • direction of flow of current

    • for a +ve charge, the current points in the same direction as its velocity

    • for a -ve charge, current points in the opposite direction to its velocity

    • velocity is same direction as motion/the force

  • direction of magnetic field

force on a charged particle moving a magnetic field

• acts perpendicular to the field and the particle’s velocity

  • as a result, it follows a circular path

origin of Hall voltage

• when an external magnetic field is applied perpendicular to the direction of current through a conductor

  • the electrons experience a magnetic force

• as a result, the electrons drift to one side of the conductor, causing it to become more negatively charged

  • this causes the opposite side to become more positively charged

• as a result of the separation of charge, a potential difference is set up across the conductor

  • this is the Hall voltage

define Hall voltage

• the potential difference produced across an electrical conductor

• when an external magnetic field is applied

• perpendicular to the current though the conductor

Hall voltage expression

V_H= Hall voltage (V)

B = magnetic flux density (T)

I = current (A)

n = number density of electrons (m^-3)

q = charge of the electron (C)

t = thickness of the conductor (m)

equation shows smaller electron density n means

• larger magnitude of Hall voltage

  • (which is why a semiconducting material is used for a Hall probe)

(1) deriving Hall voltage, V_H equation

• equation for V_H can be derived from electric and magnetic forces on the charges

  • voltage arises from electrons accumulating on one side of the conductor

  • as a result of charge separation, an electric field is set up between the two opposite sides of the conductor

(2) deriving Hall voltage

• the two sides of the conductor can be treated as oppositely charged parallel plates

  • where the electric field strength, E is equal to

  • E = V_H / d

• where V_H = Hall voltage and d = width of the conductor (m)

(3) deriving Hall voltage

• a single electron has a drift velocity of v within the conductor

  • the magnetic field is into the plane of the page, therefore the electron has magnetic force F_B to the right

    F_B = Bqv

• this is equal to the electric force F_E to the left

  • F_E = qE

• ∴ qE = Bqv

• cancel out q

  • E = Bv

  • remember E = V_H / d

  • V_H/ d = Bv

(4) deriving Hall voltage

• I is related to the drift velocity, v by

  • I = nAvq

  • where n = number density of electrons (m^-3)

  • A = cross-sectional area of the conductor (m²)

• rearrange eqn. for v

  • v = I / nAq

• substitute this for v in V_H / d = Bv

  • V_H / d = BI / nAq

(5) deriving Hall voltage

• cross-sectional area A of the slice is the product of the width of the conductor, d, and its thickness, t

  • A = dt

• substitute this into the eqn.

  V_H/ d = BI/ n(dt)q

• cancel out the d’s

• V_H = BI/ ntq

measure magnetic flux density

• using a Hall probe

  • can be used to measure the magnetic flux density between two magnets based on the Hall effect

  • consists of a cylinder with a flat surface at the end

• the flat surface of the probe must be directed between the magnets

  • to ensure the magnetic field lines pass completely perpendicular to this surface

  • the connected voltmeter to measure

    • if probe is held in wrong orientation, measured hall v will be reduced

why does Hall voltage vary when measured by Hall probe thats rotating

• Hall voltage depends on angle between angle of magnetic field and the plane of the probe

• Hall voltage reaches a maximum when field is perpendicular to the probe, and it’s zero when field is parallel to the probe

motion of a charged particle moving in a uniform magnetic field perpendicular to the direction of the motion of the particle

• when a charged particle enters a uniform magnetic field, it travels in a circular path

  • because magnetic force, F, is perpendicular to particle's velocity, v, and is directed towards the center of the circle resulting in circular motion

equate magnetic force of a charged particle moving in mag. field perpendicular to it’s motion…

• to centripetal force

F = m (v²/r) and F = BQv

therefore m (v²/r) = BQv

therefore r = mv / BQ

• where

  • r = radius of path (m)

  • m = mass of particle (kg)

  • v = linear velocity (m s^-1)

  • B = magnetic field strength (T)

  • Q = charge of the particle (C)

this equation shows that

• faster moving particles with speed v move in larger circles

  • r ∝ v

• particles with greater mass m move in move in larger circles

  • r ∝ m

• particles with greater charge q move in smaller circles

  • r ∝ 1/q

• particles moving in stronger mag. field B move in smaller circles

  • r ∝ 1/B

• also centripetal acceleration is in the same direction as the centripetal force => F = ma

velocity selector

a device consisting of perpendicular electrical and magnetic fields where charged particles with a specific velocity can be filtered

• consists of two oppositely charged parallel plates with electric field strength E between them

• there’s a uniform magnetic field with flux density B applied perpendicular to the electric field

to select particles traveling at exactly the desired speed…

• the electric force (F_E = Eq) doesn’t depend on the velocity

• but the magnetic force does (F_M = Bqv)

• ∴ to select particles traveling at exactly the desired speed v, the electric and magnetic force must ∴ be equal, in opposite directions

  • F_E = F_B

  • the resultant force = 0, particles will remain undeflected and pass straight through b.w. the plates

∴ find velocity

• equate both force equations

  • Eq = Bqv

  • cancel out q

  • E = Bv

  • make v subject

  • v = E/B

• therefore v is the speed @ which a particle will remain undeflected

  • if the speed is > or < v, then the magnetic force will deflect it and collide w one of the charged particles

  • removing particles in the beam that aren’t exactly v

• grav. force negligible, can be ignored in calculations

if velocity increases

• the magnetic force must be greater as  F_B ∝ v

magnetic fields formed in…

• wherever there’s a current flow, such as:

  • long straight wires

  • long solenoids

  • flat circular coils

magnetic fields around a current-carrying wire

• the mag. field lines are circular rings, centered on the wire

• field lines become closer near the wire, where field is strongest, and become further apart w/ distance & field becomes weaker

• reversing current reverses direction of field (clockwise/anti-clockwise)

• use right hand rule to see

  • if current is up - anticlockwise

magnetic field around a solenoid

• a solenoid increases mag. flux density B by adding more turns onto the wire

  • one end is north pole (field lines emerge), other south pole (field lines return)

• use right hand rule to determine poles

  • north pole forms where current flows anti-clockwise and vice-versa

magnetic field around a flat circular coil

• flat circular coil is equivalent to one coil around a solenoid

• field lines emerge through one side of the circle (north pole) and enter through the other (south pole)

increase magnetic field due to the current in a solenoid by…

• adding more turns to the coil

  • concentrates the magnetic field lines

• adding a ferrous core

  • when current flows through a solenoid with an iron core, it becomes magnetic, creating an even stronger field

origin of forces between current-carrying conductors

• a current-carrying conductor produces a magnetic field around it

• parallel current-carrying conductors therefore attract/repel each other

  • if currents are in same direction, magnetic field lines b.w. them cancel out - they attract each other

  • if currents are in opp. directions, magnetic field lines push each other apart - they repel each other

define magnetic flux

• the product of the magnetic flux density, B and the cross-sectional area perpendicular to the direction of the magnetic flux density

• aka measure of magnetic field lines passing through a given area

electromagnetic induction

• process of inducing e.m.f. in a conductor where there’s relative movement between a charge and a magnetic field

  • when a conductor cuts through magnetic field lines

• amt. of e.m.f. depends on magnetic flux → which varies as the coil rotates within the field

• the flux is the total magnetic field that passes through a given area

  • max. when mag. field lines are perpendicular to plane of the area, 0 when they’re parallel

formula for magnetic flux

Φ = BA

• Φ = magnetic field (Wb)

• magnetic flux density (T)

• cross-sectional area (m^-3)

when magnetic field lines aren’t completely perpendicular to the area A, the formula becomes…

Φ = BA cos θ

• θ = angle between magnetic field lines and the line perpendicular to the plane of the area

magnetic flux is max. when

• magnetic field lines are perpendicular to plane of area

• vice versa, parallel

an e.m.f. is induced when there’s

• magnetic flux linkage changes w.r.t. time

  so when there’s

  • changing B

  • changing A

  • change in angle θ

magnetic flux linkage formula/definition

• product of the magnetic flux and number of turns

• used for solenoids

• mag. flux linkage = NΦ 

• NΦ  = N(BA)

• unit Weber turns

• when field lines not perpendicular

  • NΦ = BANcosθ

cross-sectional area of solenoid

•  A = πr²

• same as a circle

electromagnetic induction

• occurs when e.m.f. is induced due to relative movement between the conductor and the magnetic field

• either conductor moves relative to a magnetic field or a magnetic field varies relative to a conductor

when a conductor cuts through magnetic field lines

• free electrons in the conductor experience a magnetic force

• causes work to be done as charges in the conductor become separated

• mechanical work is transferred to the charges as electrical potential energy

• a p.d. is created b.w. the two ends of the conductor, aka an e.m.f. is induced

experiment 1 - moving a magnet through a coil

• changing magnetic flux induces e.m.f. in a circuit

• connect coil to sensitive voltmeter, move bar magnet in and out to induce an e.m.f.

• when magnets not moving → voltmeter shows 0 reading → rate of change of flux is zero → no e.m.f. induced

• when bar magnet is moved out of the coil → current changes direction → an e.m.f. is induced in the opposite direction

• increase speed of magnet → rate of change of flux increases → e.m.f. of higher magnitude

increase magnitude of induced e.m.f.

• move the magnet faster through the coil

• add more turns to the coil

• increase strength of bar magnet

experiment 2 - moving a wire through a magnetic field

• when a long wire is connected to a voltmeter and moved between two magnets, an e.m.f. is induced (there’s no current flowing through wire to start w/)

• wire’s not moving → rate of change of flux 0 → voltmeter reads 0

• when wire’s moved back out field → direction of current changes → e.m.f. generated in opposite direction

increase magnitude of e.m.f.

• increase length of wire

• move wire faster

• increase strength of magnets

Faraday’s law of electromagnetic induction

• the magnitude of the induced e.m.f. is directly proportional to the rate of change in magnetic flux linkage

Lenz’s law

• gives direction of induced e.m.f. (as defined by Faraday’s law)

• the induced e.m.f. acts in such a direction to produce effects which oppose the change causing it

• ε = -N∆/Φ∆t

the equation shows

• when a bar magnet moves through a coil, an e.m.f. is induced within the coil due to a change in magnetic flux

• a current is also induced which means the coil now has its own magnetic field

• coil’s magnetic field acts in opp. direction to bars magnetic field

• reversing magnet direction gives opp. deflection in ammeter

• induced field in a coil repels the bar magnet

  • as direction of induced field in coil pushes against the change creating it → the bar magnet