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ECT1026 Field Theory Chapter 3 Magnetostatics 3.2 Magnetic Force By Dr Mardeni Roslee [email protected] 0383125481 1 ECT1026 Field Theory Lecture 3-2 2007/2008 Charge Particles 3.2 Magnetic Force on Current Carrying Conductor 3.3 The Biot-Savart Law 2 ECT1026 Field Theory 3.2 Magnetic Force I. Force on a Moving Charge in B Fm = qu u B (N) -q q + u A test charge q is in a magnetic field B experiences a magnetic force, Fm. This force is: proportional to q; the direction of Fm at any point is at right angle to the velocity vector, u and magnetic filed B 3 ECT1026 Field Theory The magnetic force acting on the particle of charge “q” = Fm = q u × B (N) Magnitude of Fm is the angle between u and B = 90o Fm is maximum = 0o or 180o Fm =0 Direction of Fm = direction of u B Right Hand Rule 4 ECT1026 Field Theory Right Hand Rule - Direction of Fm Fm = q u × B Thumb: direction of Fm Fingers: direction of velocity u 5 ECT1026 Field Theory 3.2 Magnetic Force 2007/2008 If a charged particle is present in both an electrical field, E and magnetic field, B. According to Lorentz Force law, the total electromagnetic force acting on the particle = F = Fe + Fm =qE+quB = q (E + u B) 6 ECT1026 Field Theory 3.2 Magnetic Force Differences between Electric Force and Magnetic Force Electric force is always in the direction of the electric field. The magnetic force is always perpendicular to the magnetic field Fe The electric force acts on a charged particle whether or not it is moving, the magnetic force acts on it only when it is in motion Fe Fm The electric force expends energy in displacing a charged particle, the magnetic force does no work when a particle is displaced V1 Fe V2 V1 Fm V2 (V1=V2) Work done (V1=V2) No Work done 7 ECT1026 Field Theory The electric force expends energy in displacing a charged particle, the magnetic force does no work when a particle is displaced Since Fm is always perpendicular to u Fm . u = 0 (Fm.U.Cos 90 = 0) The work performed when a charged particle is displaced by a (S=ut) differential distance dl = u dt is dW = Fm . dl = (Fm . u ) dt = 0 (W=Fs) 8 ECT1026 Field Theory Since no work is done, a magnetic field cannot change the kinetic energy of a charged particle Magnetic field can change the direction of motion of a charged particle, but it cannot change its speed 9 ECT1026 Field Theory 3.2 Magnetic Force II. Force on a Current Carrying Conductor A slightly flexible vertical wire is oriented along the z-direction, is placed in a magnetic field B directed into the page a) No current flowing in the wire, Fm= 0. The wire maintains its vertical orientation 10 ECT1026 Field Theory 3.2 Magnetic Force Force on a Current Carrying Conductor dFm= I dl B Fm Fm Current is flowing upward in the wire Current is flowing downward in the wire Wire deflects to the left (-y–direction) Wire deflects to the right (+y–direction) 11 ECT1026 Field Theory 3.2 Magnetic Force Force on a Current Carrying Conductor in B For a closed contour C carrying a current I, the total magnetic force is: (N) The equation is applied to two special cases: 1) Closed circuit in an uniform B field 2) Curved wire in an uniform B field 12 ECT1026 Field Theory 3.2 Magnetic Force Force on a Current Carrying Conductor in B Case 1) Closed Circuit in a Uniform B Field Consider a closed wire carrying a current I and placed in a uniform external magnetic field B. B is constant, thus it is taken outside the integral. 13 ECT1026 Field Theory 3.2 Magnetic Force Force on a Current Carrying Conductor in B Case 1) Closed Circuit in a Uniform B Field a The integral of vectors dl over a closed path is equal to zero. Therefore, the total magnetic force on any closed current loop in a uniform magnetic field is zero. (a=a) 14 ECT1026 Field Theory b = a b 3.2 Magnetic Force 2007/2008 Force on a Current Carrying Conductor in B Case 2) Curved Wire in a Uniform B Field Consider a curved wire carrying a current I and placed in a uniform external magnetic field B. B is constant, thus it is taken outside the integral. a Fm = I (a = b) (∫adl) B = I l B b l is the vector directed from a to b, integral of dl from a to b has the same value irrespective of the path from point a to b 15 ECT1026 Field Theory 3.2 Magnetic Force Force on a Current Carrying Conductor in B Case 2) Curved Wire in a Uniform B Field The net magnetic force on a line segment is proportional to the vector between the end points (point a and b), 16 Coulomb’s Law dq=vΔv 17 ECT1026 Field Theory 3.3 The Biot-Savart Law Point R B Rr I dl The magnetic flux density at a point R from an element dl of a wire that is carrying a current I. “dB is generated by I” 18 ECT1026 Field Theory 3.3 The Biot-Savart Law The magnetic flux density at a point R from an element dl of a wire that is carrying a current I. This expression is known as the Biot-Savart law, and it is expressed as: ^ o0 IIddl×R Rˆ dB dB 2 2 4 4 RR (Tesla) It is also defined as the differential magnetic flux density dB generated by a steady current I flowing through a differential length dl 19 ECT1026 Field Theory 3.3 The Biot-Savart Law (Observation point) Difference between E and B Electric field vector E, whose direction is along the distance vector joining the ‘charge’ and ‘observation point’. q ^ E=R 4eR2 The magnetic field or magnetic flux density is orthogonal to the plane containing dl and R. (the direction is dictated by dl R Point R B B R Rr dl E I (Charge) dl B U Fm Hence, B is directing “out of the page” by using right hand rule. 20 ECT1026 Field Theory 3.3 The Biot-Savart Law Surface Current Distribution o B s 4 Js × Rˆ R2 Volume Current Distribution (I dl = Js ds = Jdv) ds Js = Surface Current Density o B v 4 J × Rˆ R2 J = Volume Current Density dv 21