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Electric Field due to a Point Charge ౼ Problems and Solutions

1. Two particles with charges +4μC and -9μC are kept fixed at a separation of 20 cm from each other. Locate the point at which the resultant electric field due to the system of two point charges is zero.

Here the question asks to find the null point where the resultant electric field due to the given system of point charges is zero. We know that the null point, in case of a system of two point charges, lies on the line joining the two point charges and also the null point always lies closer to the smaller magnitude charge. Since the two charges are of opposite nature, the null point will not lie in the space between the charges.
Let at point P the electric field is zero. The point P is at a distance say x from the +4μC charge as shown in the diagram. At point P, the field vanishes and therefore the magnitudes of individual fields produced by the two point charges at point P must be equal (and directions anti-parallel). 
We get two values of x; +40 cm and -8 cm. The negative value of x should be neglected as there isn't any possibility of the null point in the region between the two charges.
Hence the point where the electric field is zero, lies on the line connecting the two point charges and at a distance of 40 cm on the left side of the positively charged particle.

2. A point charge -2μC is located at point A(2,2,2), then find the electric field strength vector at point B(1,1,1).

Electric field intensity vector due to a point charge q at a position from it is expressed by the formula 
Here r is the position vector of point of calculation of electric field with respect to the location point of the source point charge and q is to be put with proper sign. 
which is the sought field vector at point B.

3. Two identical particles each having charge +q are situated at points A(-√2,0) and B(√2,0). At which point(s) of y-axis the electrical field intensity is maximum?
Let P(0,y) be a point lying on the y-axis. We should first proceed to get the value of field intensity at the very point P.
Let the field produced by each charged particle at point P be E. Then we can write
This is the expression of net electric field at an arbitrarily selected point P on the y-axis. Now for field on a particular point(s) of y-axis to be maximum, its derivative with respect to the y coordinate should be equal to zero. 
Therefore the points of maximum field intensity on y-axis are (1,0) and (-1,0). These two points are symmetrically located with respect to the x-axis.