Flux Density due to a current flowing in a long straight wire © David Hoult 2009

The field at point p is directed © David Hoult 2009

The field at point p is directed out of the plane of the diagram (“corkscrew rule”) © David Hoult 2009

The magnitude of B at point p depends on © David Hoult 2009

The magnitude of B at point p depends on the current, I © David Hoult 2009

The magnitude of B at point p depends on the current, I the perpendicular distance of p from the wire © David Hoult 2009

The magnitude of B at point p depends on the current, I the perpendicular distance of p from the wire the medium surrounding the wire © David Hoult 2009

Experiments show that B  I and if r is small compared with the length of the wire then © David Hoult 2009

and if r is small compared with the length of the wire then Experiments show that Therefore B  I B  1 r © David Hoult 2009

B  I and if r is small compared with the length of the wire then Experiments show that B  1 r Therefore B = I r (a constant) © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r where µ is the permeability of the medium surrounding the wire © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r where µ is the permeability of the medium surrounding the wire If the medium is a vacuum (or air) the permeability is written as µ o © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r where µ is the permeability of the medium surrounding the wire If the medium is a vacuum (or air) the permeability is written as µ o The units of µ are © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r where µ is the permeability of the medium surrounding the wire If the medium is a vacuum (or air) the permeability is written as µ o The units of µ are T A -1 m -1 = © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r where µ is the permeability of the medium surrounding the wire If the medium is a vacuum (or air) the permeability is written as µ o The units of µ are T A -1 m -1 = NA -2 © David Hoult 2009

Because this is a situation having cylindrical symmetry, the factor 2  is included in the equation B = µ Iµ I 2 r2 r where µ is the permeability of the medium surrounding the wire If the medium is a vacuum (or air) the permeability is written as µ o The units of µ are T A -1 m -1 = NA -2 1 N A -2 = 1 Henry per meter (H m -1 ) © David Hoult 2009

Force acting between two long, parallel, current- carrying conductors © David Hoult 2009

Current I 2 flows through the field produced by current I 1 (and vice versa) © David Hoult 2009

Current I 2 flows through the field produced by current I 1 (and vice versa) Flux density near conductor 2 produced by I 1 is given by © David Hoult 2009

Current I 2 flows through the field produced by current I 1 (and vice versa) B = µo I1µo I1 2 r2 r Flux density near conductor 2 produced by I 1 is given by assuming that the medium is a vacuum (or air) © David Hoult 2009

Force acting on a length L of wire 2 is F = I 2 L B © David Hoult 2009

Force acting on a length L of wire 2 is F = I 2 L B Therefore, force per unit length acting on wire 2 is © David Hoult 2009

Force acting on a length L of wire 2 is F = I 2 L B Therefore, force per unit length acting on wire 2 is µ o I 1 I 2 2 r2 r F L = © David Hoult 2009

µ o I 1 I 2 2 r2 r F L = 1 A is the current which, © David Hoult 2009

µ o I 1 I 2 2 r2 r F L = 1 A is the current which, when flowing in each of two infinitely long, straight, parallel conductors, © David Hoult 2009

µ o I 1 I 2 2 r2 r F L = 1 A is the current which, when flowing in each of two infinitely long, straight, parallel conductors, separated by 1m, © David Hoult 2009

µ o I 1 I 2 2 r2 r F L = 1 A is the current which, when flowing in each of two infinitely long, straight, parallel conductors, separated by 1m, in a vacuum, © David Hoult 2009

µ o I 1 I 2 2 r2 r F L = 1 A is the current which, when flowing in each of two infinitely long, straight, parallel conductors, separated by 1m, in a vacuum, produces a force per unit length of 2 × N m -1 © David Hoult 2009

µ o I 1 I 2 2 r2 r F L = 1 A is the current which, when flowing in each of two infinitely long, straight, parallel conductors, separated by 1m, in a vacuum, produces a force per unit length of 2 × N m -1 © David Hoult 2009

Flux density produced by a long coil (solenoid) Current flowing through a conductor produces a magnetic field. If the conductor is a long straight wire, then the field is distributed over a large region of space. If the wire is used to make a coil, the magnetic field is concentrated into a smaller space and is therefore stronger © David Hoult 2009

The flux density, B c at the centre of a long coil, having N turns and of length L depends on © David Hoult 2009

the current flowing through the solenoid, I The flux density, B c at the centre of a long coil, having N turns and of length L depends on © David Hoult 2009

the current flowing through the solenoid, I The flux density, B c at the centre of a long coil, having N turns and of length L depends on the number of turns per unit length © David Hoult 2009

the current flowing through the solenoid, I The flux density, B c at the centre of a long coil, having N turns and of length L depends on the number of turns per unit length the permeability of the medium inside the solenoid © David Hoult 2009

Experiments show that the flux density, B c on the axis, at the centre of a solenoid is directly proportional to I directly proportional to N/L © David Hoult 2009

B c  I NI N L © David Hoult 2009

B c  I NI N L The constant of proportionality is µ (the permeability of the medium), therefore we have © David Hoult 2009

B c  I NI N L The constant of proportionality is µ (the permeability of the medium), therefore we have B c = µ I Nµ I N L © David Hoult 2009

The flux density on the axis at the end of the solenoid is equal to © David Hoult 2009

The flux density on the axis at the end of the solenoid is equal to B c / 2 © David Hoult 2009

The flux density on the axis at the end of the solenoid is equal to B c / 2 © David Hoult 2009

The flux density on the axis at the end of the solenoid is equal to B c / 2 © David Hoult 2009