Physics as a hobby

Q. The velocity of a solid object rotationg about an axis is a field . Show that a) b) where = angular velocity. Trial. a) If velocity has radial component or magnitude of angular velocity is not constant, solid object should be broken, so velocity has no radial component and magnitude of angular velocity is constant. Let’s take coordinate such that solid object is rotating in plane, axis is and use for magnitude of angular velocity. In cylinderical coordinate, we can write , . Then Divide by Radial componet of velocity is zero and , so from eq(3), (4) we get Using eq(5),(6) and (7), eq(8) can be wrriten b) Suppose rotation is counter clock wise. Then we can write Curl is Using eq(5) & (6) and (10), eq(11) can be written

Q. A copper wire of radius has a concentric insulating sheath with outer radius . The wire carries an electric current that raises its temperature to while the outside of the insulation remains at near room temperature. a) What is inside the insulation? Give answer in terms of , , , . b) How big is the temperature difference if a current of 20 amps is sent through No. 10 gauage copper wire which is covered with a layer of rubber 0.2 cm thick whose thermal conductivity is ? Trial. a) b) Copper wire generate heat by Joule heating [ 1 ] Let length of copper wire , and cross-sectional area , then , where is electrical resistivity [ 2 ] . From eq(2) and (3) Radius is , Putting eq.(5) to eq(4) gives This heat trasfered throgh rubber layer, will be , where is area heat transfered, (“-” means lose energy). Suppose , then If proportionality constant is , from (7) & (8) Using eq(6), (9), mean values No...

Solenoidal If the is zero, is solenoidal , then is the curl of some vector filed . Theorem If there is a such that Irrotational If the is zero, is irrotational , then is the gradient of some scalar field . Theorem If there is a such that

Q. A particle with a mass and a positive charge is at a point , , and is moving with a low velocity The charge is influenced by a negative charge fixed at the origin and by a uniform magnetic field in the direction. How large must be such that the moving particle describes a circle of radius about the stationary one? If the magnitude of the magnetic field strength were different than this, explain why the speed of the particle is a function of the radial distance only. Sketch qualitatively several cycles of the trajectory followed by the particle if it were released from the point , with zero velocity. Trial. For circular motion, For our system, if eq(1) is satisfied, it starts circular motion, then at start time radius , and From eq(1), From eq(4), From eq(5) & eq(6), condition of for circular motion is Motion is always in x-y plane, because force is always toward origin. That the force is always toward origin means is influenced only by radia...

Q. In the region of space of interest there is a uniform magnetic field, such that , , . The field is constant in time, and there are no currents or electric fields in the region of space we consider. A particle of mass and positive charge is started at , , with a velocity in the direction. Sketch and describe quantitatively in terms of , , , and the path of the particle. (Assume ) Suppose, that , , but . For always small compared to , but not completely negligible, show on a sketch the qualitative behavior of the particle trajectory. (See Charpak, et al, Physical Review Letters, Vol. 6, 128 (1961) for the use of a similar field in an important expriment.) Show that the field just postulated is inconsistent with one of Maxwell’s equations if the field fills a finite volume of space and, as above, you assume there are no currents or electric fields in the volume. Trial. For , , , From Initial condition, , , Trajectory is hyperbola. If , from eq.(...

Q. In a certain region of space, there is a uniform electric field of 10,000 volts per centimeter in the direction. There is in the same region a uniform magnetic filed in the direction. A beam of mu-measons with the velocity travels through this region on a straight line in the direction. a) What is the strength of the field ? (A mu-meason has a mass 210 times the electron mass and a charge equal in magnitude to the electron charge.) b) Can you tell from this experiment if the charge on the measons is + or -? Trial. Electromagnetic force a) From eq.(1) , force is . Problem said muons move on a straight line with velocity , which means no external force , . b) I can’t tell muon is + or - charge only with this experiment.