EJC Physics H216 Electromagnetism -1. Notes (2024) for print

Magnetically levitated (maglev) trains exploit the physics of electromagnetism to (i) lift a train off the tracks to minimize friction and (ii) accelerate or perform regenerative braking of the train. Content • Concept of a magnetic field • Magnetic fields due to currents • Force on a current-carrying conductor • Force between current-carrying conductors • Force on a moving charge Learning Objectives: Candidates should be able to: (a) show an understanding that a magnetic field is an example of a field of force produced either by current -carrying conductors or by permanent magnets (b) sketch flux patterns due to currents in a long straight wire, a flat circular coil and a long solenoid (c) use 00 0 an 2 2 d BB ,d Br  = = =I NI nI for flux densities of the fields due to currents in a long straight wire, a flat circular coil and a long solenoid respectively (d) show an understanding that the magnetic field due to a solenoid may be influenced by the presence of a ferrous core (e) show an understanding that a current-carrying conductor placed in a magnetic field might experience a force (f) recall and solve problems using the equation sin FB = Il , with directions as interpreted by Fleming's left -hand rule (g) define magnetic flux density (h) show an understanding of how the force on a current-carrying conductor can be used to measure the flux density of a magnetic field using a current balance (i) explain the forces between current-carrying conductors and predict the direction of the forces (j) predict the direction of the force on a charge moving in a magnetic field (k) recall and solve problems using sin F BQv = (l) describe and analyse deflections of beams of charged particles by uniform electric and uniform magnetic fields (m) explain how electric and magnetic fields can be used in velocity selection for charged particles.

Earlier in gravitational fields and electric fields, we described the directions along which bodies will experience forces, and quantitatively found the field strengths. Back at the secondary level we could deduce the direction of force that will act on a current-carrying conductor subject to a magnetic field. At the A-Levels, we will go on further to quantify how strong a magnetic field is. Curiously the definition reads as “ may experience a force”; recall that if the direction of current is parallel to the magnetic field lines, there will be no force due to the magnetic field. Strength of a magnetic field is represented by a vector quantity called the magnetic flux density B. uniform magnetic field non-uniform magnetic field Parallel and equally -spaced lines. Number of lines passing through an area (for comparison) is the same. As with gravitational fields and electric fields, we represent magnetic fields using magnetic field lines. Field lines do not intersect. When two or more fields interact, the resultant is a vector addition. unit cross-sectional area B stronger field weaker field B magnetic field lines: • come out of north poles and go into south poles • (the tangent) shows the direction of force that a ‘free’ magnetic north pole will experience at that point • are closer together where the field is stronger A magnetic field is a region of space in which a permanent magnet, a current-carrying conductor or a moving charge may experience a force

A compass needle will point in the field direction at that point A compass needle aligns itself with the Earth’s magnetic field . Therefore, the geographic North pole is a magnetic South pole ; only then can the North -pointing ends of compass needles point in the “correct” direction. We use dots and crosses to represent magnetic field lines going into and out of the plane of paper. uniform magnetic field pointing into plane of paper non-uniform magnetic field pointing out of plane of paper When using the right-hand grip rule, the 4 fingers can be either the flow of current or the direction of the magnetic field lines: four fingers give direction of magnetic field lines four fingers aligned to direction of current flow There are three geometries you need to be familiar with: 1. Magnetic flux density due to current in a long straight wire 2. Magnetic flux density due to a flat circular coil 3. Magnetic flux density due to a long solenoid plane of paper × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × I B I B

3D view top view side view The magnetic field lines are concentric circles centred on the wire. The radius of each larger circle is increasingly larger because the magnetic field is weaker further away from the wire. The magnetic flux density (B) due to a long straight wire at a perpendicular distance d away from the wire carrying current I is given in the list of data and formulae. You have to remember what the individual quantities refer to. 0 : permeability of free space (H m-1) I : current in the conductor (A) d : distance from the conductor (m) B I d 1. 5 1 1 B I d × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × × I B d 0 2B d  = I

Example 1 A vertical wire X carries a current of 5.0 A downwards. (a) Find the magnetic flux density due to the current at point P that is 10 cm east of X. (b) The horizontal magnetic flux density due to Earth’s magnetic field is 40 T . Find the resultant magnetic flux density at P. (c) Determine the location relative to X where the resultant magnetic flux density is zero. (d) State if the resultant magnetic flux density 10 cm North of X is greater than at P. Solution ( )( ) ( ) ( ) ( ) 7 0 wire 2 5 net Earth wire 65 5 net Earth wire 60 new new 10 (a) 10 10 1 0 10 T towards South (b) 10 1 0 10 3 0 10 T towards North (c) 10 0 4 5 0 2 2 40 0 025 m 42 0 B d . B B B . . B B B d d . .      − − − −− − −  =  =  −  = = = = =− = = − = I I East of X (d) larger in magnitude. magnetic flux density of Earth vectorially added to magnetic flux density due to current carrying wire. 3D view top view, current-carrying coil top view The magnetic flux density at the centre of a flat circular coil with radius r, number of turns N, carrying current I is given in the list of data and formulae. We need to know what the individual quantities refer to. 0 : permeability of free space (H m-1) N : number of turns I : current in the conductor (A) r : radius of coil (m) I B r plane of paper B r B I I r 0 2B N r = I Bwire I 10 cm Top view N BEarth Bwire BEarth I Bnet

3D view top view, current-carrying coil top view A long solenoid is one where its length is significantly larger than its (cylindrical) radius. The magnetic field lines are mostly parallel, straight and equally-spaced in the centre region of the long solenoid - the field is more uniform towards the centre of the coil. The field lines begin to diverge nearer the ends of the solenoid. The magnetic flux density along the axis of a long solenoid that has n number of turns of wire per unit length is given in the list of data and formulae. We need to know what the individual quantities refer to. 0 : permeability of free space (H m-1) n : number of turns per unit length I : current in the conductor (A) Note that 0 is the permeability of free space, a constant that describes how magnetic fields behave in free space (or vacuum), similar to how permittivity of free space 0 describes the behaviour of electric fields. The equation is not valid if a ferrous (e.g. soft iron) core is inside the solenoid. I B I B n (per metre) B n (per metre) I Question: Explain the effects