FI 2201 Electromagnetism

Transcription

1 FI 2201 Electromagnetism Alexander A. Iskandar, Ph.D. Physics of Magnetism and Photonics Research Group Magnetostatics CURRENT AND MAGNETIC FIELDS 1

2 Current Consider a long conducting wire that is neutral overall but contains charges that can move under the influence of an externally-applied electric field, and charges that can t: + E r λ 0 λ λ The motion of the +λ distribution is a steady current: I λv : Coulomb/second Ampere During a time t a charge Q λv t passes a given point on the wire; I is the rate at which charge passes that point, I Q/ t. 3 v r Current Though in general v is of course a vector, we don t usually write I as a vector (currents through wires, one dimension). However, it is useful to think of the surface and volume analogues of this line-charge current as vectors, since they will come into any discussion of currents within a wire. Suppose we have a conducting sheet, with surface density σ in mobile charges and σ in fixed charges. Then the surface current density is K σ v : Ampere/m The total current from the sheet r is l di I Kdl K dl where dl is an infinitesimal length element perpendicular to the direction the current flows. 4 2

3 Current Suppose you have a conducting volume, with charge density ρ of mobile charges and ρ of fixed charges. Then we get to define the most useful of current densities, the volume current density : J ρ v : Ampere/m 2 r a r r di I Jda J da J da where a is an infinitesimal it i area element perpendicular to the direction the current flows. 5 Continuity Equation Integrating over the whole surface area of a conductor, one gets the current flowing out of the volume it encloses: I J da J dτ S Because electric charge is conserved, the only way for a current to flow out of a volume is for the charge inside it to decrease: dq ρ r I dτ J dτ dt t V V or r ρ J + 0 t the continuity equation. V 6 3

4 Consider setting up the following experiments + + We observe that the wires are repelling each other in case 1 and attracting each other in case 2. 7 The conducting wire is neutral overall but contains charges that can move under the influence of an externally-applied electric field, and charges that can t: + E r λ 0 λ λ However, although we have positive charge density and negative charge density, but still the previous observation of attraction and repelling can not be explained. We encounter a new phenomena, hence we need a new physical quantity to explain this phenomena. 8 v r 4

5 Moving charges produces a magnetic field B r in the space around it, in addition to the electric field E r. This observation of magnetic field produced by an electrical current was first demonstrated by Ørsted. Later on in this chapter, we will learn how to calculate the magnetic field produced by this current. The term magnetostatics means that the magnetic field does not change with time. 9 The magnetic force on a charge Q moving with velocity in a magnetic field B r v r is F Q v B When both electric and magnetic fields are present, the charge Q will experience the Lorentz force r F QE + Q v B The magnetic field has a unit of Tesla 10 4 Gauss Example

6 The magnetic force do no work on a charge Q. If the charge move a distance dl vdt then the work by magnetic force will be dw F dl Q v B vdt ( ) 0 11 The magnetic force on a current can be calculated as follows r F ( v B ) dq ( v B ) λ dl ( I B ) dl However, for a constant current in one dimensional problem, the direction of I r and dl r are the same, then we can write F I ( dl B) I ( dl B) Likewise magnetic force on surface current density and volume current density are F ( K B) da F ( J B) dτ 12 6

7 Example A rectangular circuit, lying in the y-z plane, carries a current I and lies partly in a region where there is a constant magnetic field in the x direction. A weight, mass m, hangs from the circuit. a. What is I, given m? b. What happens if I is made larger than the value found in a? 13 The magnetic force is Fmag I dl B b a 0 I ẑ xˆb dz ŷ xˆb dy ẑ xˆb dz b IaBẑ mg W so mg I ab The loop will rise until its lower edge enters the field region. (Don t get the impression from this that the magnetic field is doing work to lift the weight; the work is done by the battery.) 14 7