May 2011 - Career Point

the indivual forces. It's often helpful to usecomponents in an xy-coordinate system. Be sureto use correct vector notation; if a symbolrepresents a vector quantity, put an arrow over it.If you get sloppy with your notation, you will alsoget sloppy with your thinking.As always, using consistent units is essential.With the value of k = 1/4πε 0 given above,distances must be in meters, charge in coulombs,and force in newtons. If you are given distance incentimeters, inches, or furlongs, donot forget toconvert ! When a charge is given inmicrocoulombs (µC) or nanocoulombs (nC),remember that 1µC = 10 –6 C and 1nC = 10 –9 C.Some example and problems in this and laterchapters involve a continuous distribution ofcharge along a line or over a surface. In thesecases the vector sum described in Step 3 becomesa vector integral, usually carried out by use ofcomponents. We divide the total chargedistribution into infinitesimal pieces, useCoulomb's law for each piece, and then integrateto find the vector sum. Sometimes this processcan be done without explicit use of integration.In many situations the charge distribution will besymmetrical. For example, you might be asked tofind the force on a charge Q in the presence oftwo other identical charges q, one above and tothe left of Q and the other below and to the left ofQ. If the distance from Q to each of the othercharges are the same, the force on Q from eachcharge has the same magnitude; if each forcevector makes the same angle with the horizontalaxis, adding these vectors to find the net force isparticularly easy. Whenever possible, exploit anysymmetries to simplify the problem-solvingprocess.Step 4 : your answer : Check whether your numericalresults are reasonable, and confirm that the likecharges repel opposite charges attract.Problem solving strategy : Electric-field calculationsStep 1: the relevant concepts : Use the principle ofsuperposition whenever you need to calculate theelectric field due to a charge distribution (two ormore point charges, a distribution over a line, surface,or volume or a combination of these).Step 2: The problem using the following steps :Make a drawing that clearly shows the locationsof the charges and your choice of coordinate axes.On your drawing, indicate the position of the fieldpoint (the point at which you want to calculate theelectric field E r ). Sometimes the field point willbe at some arbitrary position along a line. Forexample, you may be asked to find E r at point onthe x-axis.Step 3 : The solution as follows :Be sure to use a consistent set of units. Distancesmust be in meters and charge must be incoulombs. If you are given centimeters ornanocoulombs, do not forget to convert.When adding up the electric fields caused bydifferent parts of the charge distribution,remember that electric field is a vector, so youmust use vector addition. Don't simply addtogether the magnitude of the individual fields:the directions are important, too.Take advantage of any symmetries in the chargedistribution. For example, if a positive charge anda negative charge of equal magnitude are placedsymmetrically with respect to the field point, theyproduce electric fields of the same magnitude butwith mirror-image directions. Exploiting thesesymmetries will simplify your calculations.Must often you will use components to computevector sums. Use proper vector notation;distinguish carefully between scalars, vectors, andcomponents of vectors. Be certain thecomponents are consistent with your choice ofcoordinate axes.In working out the directions of E rvectors, becareful to distinguish between the source pointand the field point. The field produced by a pointcharge always points from source point to fieldpoint if the charge is positive; it points in theopposite direction if the charge is negative.In some situations you will have a continuousdistribution of charge along a line, over a surface,or through a volume. Then you must define asmall element of charge that can be considered asa point, finds of all charge elements. Usually it iseasiest to do this for each component of E rseparately, and often you will need to evaluateone or more integrals. Make certain the limits onyour integrals are correct; especially when thesituation has symmetry, make sure you don'tcount the charge twice.Step 4 : your answer : Check that the direction of E ris reason able. If your result for the electric-fieldmagnitude E is a function of position (say, thecoordinate x), check your result in any limits forwhich you know what the magnitude should be.When possible, check your answer by calculating itin a different way.Problem solving strategy : Gauss's LawStep 1 : Identify the relevant concepts : Gauss's lawis most useful in situations where the chargedistribution has spherical or cylindrical symmetry oris distributed uniform over a plane. In these situationswe determine the direction of E r from the symmetryof the charge distribution. If we are given the chargedistribution. we can use Gauss's law to find the themagnitude of E r . Alternatively, if we are given thefield, we can use Gauss's law to determine the detailsXtraEdge for IIT-JEE 22 MAY 2011