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in this example we are given that a point charge q is moving with speed v toward a fixed
earthed metal sphere of radius-a , as we can see here , and we are required to find the
current through the ammeter when charge is at a distance 3-a, from centre of sphere,
obviously as the charge is moving the distance is varying so when x equal to 3a we are required
to find the current through the ammeter. now in this situation we know that due to
this charge q there will be some potential of the sphere, and as it is connected to earth
, earth will supply a charge q-e to it and, as the charge is getting close the magnitude
of charge q-e will continuously change, to keep the potential of sphere, at zero. so
here we can write in solution , as earth is spherical, we can write that potential of
sphere is always equal to zero. so when the charge is at a distance x we can
write the potential of sphere, due to this charge will be k-q by x , + due to the charge
which is supplied by the earth , it will be k-q-e by (a) , and this should be zero, in
this situation k cancels out and the value of , charge appearing on the sphere , by the
earth will be , negative of q-a by x, now as, x is continuously changing the magnitude
of q-e will continuously change which will lead to a continuous current in the ammeter
. and here we can write , the current in ammeter
. should be equal to, d-q-e by d t because it will be the current , the rate at which
the charge on the sphere is changing continuously , so we differentiate this expression , this
will be minus q-a and the derivative of 1 by x is minus 1 by x square d x by d t, and
. here we can see this d x by d t can be written
as negative of v, or in magnitude of current we can write , q-a by x square multiplied
by v that is the magnitude of current through the ammeter , and we need to find value of
current when x is equal to 3a , so on substituting the value of, x equal to 3a here , (a) cancels
out and we get value of current is, q-a-v , by 9-a square ,a- cancels out it is q v
by 9a, this is the answer to this problem.