Solution - Career Point

If the body loses this heat in time t 1 , the temperatureat t 1 will beθ1 − θθ 1 – 0 θ1 + θ= 02 2Putting these values of time and temperature in (1) :θ1 + θ 0– θ 0 = (θ 1 – θ 0 ) e −Kt 12Ktor, e − 1 log 21= or t1 = 2 K5. In a certain region surrounding the origin of thecoordinates, → B = 5 × 10 –4 → k T and → E = kˆ V/m. Aproton enters the fields at the origin with an initialvelocity v → 0 = 2.5 × 10 5 î m/s. Describe the proton'smotion and give its position after three completerevolutions.Sol. The z-component of the force → F is a constantelectrical force. It produces a constant accelerationalong z-axis given as1Z = at 2 1 ⎛ eE ⎞= ⎜ ⎟ t22 2 ⎝ m ⎠E z ByThe other component of the force F is a magneticforce which provides the necessary centripetal forceto keep the proton in a circular path of radius r(say).The period of revolution of the proton2πrmv 20T = As F cp = = ev 0 Bv 0r∴ v 0 =eBrmHence, T =x2πmeBSince the particle (proton) moves in circular pathhaving a period of revolution T in x, y plane andmoves along z-axis with a constant accelerationa = eE/m, the path of the proton is a helix.After three revolutions, putting t = 3T, we obtain1 ⎛ eE ⎞z = ⎜ ⎟ (3T) 2 9 eET=2 ⎝ m ⎠ 2 mPutting T = 2πm/eB, we get218π Em 18×(22 / 7)z = =2−eB 1.610= 37m226× 5×1.66×10× (5×10−4)2−27Interesting Science Facts• The Universe contains over 100 billion galaxies.• Wounds infested with maggots heal quickly andwithout spread of gangrene or other infection.• More germs are transferred shaking hands thankissing.• The longest glacier in Antarctica, the Almbertglacier, is 250 miles long and 40 miles wide.• The fastest speed a falling raindrop can hit you is18mph.• A salmon-rich, low cholesterol diet means thatInuits rarely suffer from heart disease.• Inbreeding causes 3 out of every 10 Dalmationdogs to suffer from hearing disability.• The world’s smallest winged insect, the Tanzanianparasitic wasp, is smaller than the eye of ahousefly.• If the Sun were the size of a beach ball thenJupiter would be the size of a golf ball and theEarth would be as small as a pea.• It would take over an hour for a heavy object tosink 6.7 miles down to the deepest part of theocean.• There are more living organisms on the skin ofeach human than there are humans on thesurface of the earth.• The grey whale migrates 12,500 miles from theArtic to Mexico and back every year.• Quasars emit more energy than 100 giant galaxies.• Quasars are the most distant objects in theUniverse.• The Saturn V rocket which carried man to theMoon develops power equivalent to fifty 747jumbo jets.• Koalas sleep an average of 22 hours a day, twohours more than the sloth.• Light would take .13 seconds to travel around theEarth.• Neutron stars are so dense that a teaspoonfulwould weigh more than all the people on Earth.• One in every 2000 babies is born with a tooth.• Every hour the Universe expands by a billionmiles in all directions.• Somewhere in the flicker of a badly tuned TV setis the background radiation from the Big Bang.• The temperature in Antarctica plummets as lowas -35 degrees Celsius.• Space debris travels through space at over 18,000mph.XtraEdge for IIT-JEE 22 APRIL 2010