A long solenoidal cable of length l and radius a is composed of Nthin wires that each carry a steady current I. The wires are woundwith a pitch angle of theta with respect to the cable axis. Neglectthe radius of the individual wires. If the pitch is theta =0, thewires are oriented parallel to the cable axis, whereas iftheta=pi/2 the wires form circular loops in planes perpendicular tothe axis.
a) Use Ampere´s law to find the magnetic field both inside B[in]and outside B[out] of the cable. Be careful to include sketches ofthe integration contours and the vectors involved in your solution.Your answer should beB[in]=Mu_0NISin(squared)theta/2*pi*a*cos(theta),B[out]=Mu_0NIcos(theta)/2*pi*s(s=radial distance from cable axis.You must determine the direction of these fields. Hint:You cannotdirectly use the usual expression for the magnetic field of asolenoid B=Mu_onI, since the current flows at an angle to the cableaxis. Think about how to represent this current flow as twocomponent flows, each of which you can find the field for. You willalso have to determine what the turns per unit length n are. To dothis you may find it helpful to consider the distance, along thecable axis, b/w the beginning and end on one complete turn of s asingle strand. Notice that if theta=pi/2, this distance will bezero since the turn is contained entirely in a plane. What wouldthe distance be if theta=pi/2?
b)Taking a rectangular patch of the cable. Let the area of thispatch be dS. Use the maxwell stress tensor to find the force onthis patch.
c) For what pitch angle does the force on the cable vanish? Forwhich range of angles will the cable experience a ' positivepressure' acting to expand the cable diameter? For which range willthe opposite occur, i.e, a 'negative presssure 'acting to squeezethe cable?
d) If the current in the cable is switched off, what is the changein the field angular-momentum?
A long solenoidal cable of length l and radius a is composed of Nthin wires that each carry a steady current I. The wires are woundwith a pitch angle of theta with respect to the cable axis. Neglectthe radius of the individual wires. If the pitch is theta =0, thewires are oriented parallel to the cable axis, whereas iftheta=pi/2 the wires form circular loops in planes perpendicular tothe axis.
a) Use Ampere´s law to find the magnetic field both inside B[in]and outside B[out] of the cable. Be careful to include sketches ofthe integration contours and the vectors involved in your solution.Your answer should beB[in]=Mu_0NISin(squared)theta/2*pi*a*cos(theta),B[out]=Mu_0NIcos(theta)/2*pi*s(s=radial distance from cable axis.You must determine the direction of these fields. Hint:You cannotdirectly use the usual expression for the magnetic field of asolenoid B=Mu_onI, since the current flows at an angle to the cableaxis. Think about how to represent this current flow as twocomponent flows, each of which you can find the field for. You willalso have to determine what the turns per unit length n are. To dothis you may find it helpful to consider the distance, along thecable axis, b/w the beginning and end on one complete turn of s asingle strand. Notice that if theta=pi/2, this distance will bezero since the turn is contained entirely in a plane. What wouldthe distance be if theta=pi/2?
b)Taking a rectangular patch of the cable. Let the area of thispatch be dS. Use the maxwell stress tensor to find the force onthis patch.
c) For what pitch angle does the force on the cable vanish? Forwhich range of angles will the cable experience a ' positivepressure' acting to expand the cable diameter? For which range willthe opposite occur, i.e, a 'negative presssure 'acting to squeezethe cable?
d) If the current in the cable is switched off, what is the changein the field angular-momentum?
