Advertisements
Advertisements
Question
Consider a system of n charges q1, q2, ... qn with position vectors `vecr_1,vecr_2,vecr_3,...... vecr_n`relative to some origin 'O'. Deduce the expression for the net electric field`vec E` at a point P with position vector `vecr_p,`due to this system of charges.
Advertisements
Solution
Let us consider a system of n charges q1, q2, ... qn with position vectors r1, r2, r3, ...rn relative to origin O.
Let `vecF_i` be the force due to ith charge qi on q0.
Then,
`vecF_i = 1/(4πε_0) (q_1q_0)/(r_i^2) \ hat r_i`
Here, ri is the distance of the test charge q0 from qi.
The electric field at the observation point P is given by `vecE_i = lim_(q0->0) vecF_i/(q_0) = lim _(q0->0) 1/q_0 (1/(4πε_0) (q_1q_0)/r_1^2 hatr_i)`
`vecE_i = 1/(4πε_0) q_i/(r_1^2)hat r_i ........ (1) `
If `vecE` is the electric field at point P due to the system of charges, then by the principle of superposition of electric fields,
`vecE = vecE_1 +vecE_2 +vecE_3 +....vecE_n = sum_(i=1)^n vecE_i`
Using (1), we get
`vecE = sum_(i=1)^n 1/(4πε_0)q_i/(r_i^2)hat r_i `
`vecE =1/(4πε_0) sum_(i=1)^n q_i/(r_i^2)hat r_i`
APPEARS IN
RELATED QUESTIONS
A hollow cylindrical box of length 0.5 m and area of cross-section 25 cm2 is placed in a three dimensional coordinate system as shown in the figure. The electric field in the region is given by `vecE = 20 xhati` where E is NC−1 and x is in metres. Find
(i) Net flux through the cylinder.
(ii) Charge enclosed by the cylinder.
A point charge q is rotated along a circle in an electric field generated by another point charge Q. The work done by the electric field on the rotating charge in one complete revolution is
The electric field in a region is directed outward and is proportional to the distance rfrom the origin. Taking the electric potential at the origin to be zero,
Consider a uniformly charged ring of radius R. Find the point on the axis where the electric field is maximum.
A wire is bent in the form of a regular hexagon and a total charge q is distributed uniformly on it. What is the electric field at the centre? You may answer this part without making any numerical calculations.
A block of mass m with a charge q is placed on a smooth horizontal table and is connected to a wall through an unstressed spring of spring constant k, as shown in the figure. A horizontal electric field E, parallel to the spring, is switched on. Find the amplitude of the resulting SHM of the block. 
Assume that each atom in a copper wire contributes one free electron. Estimate the number of free electrons in a copper wire of mass 6.4 g (take the atomic weight of copper to be 64 g mol−1).
When 1014 electrons are removed from a neutral metal sphere, the charge on the sphere becomes ______.
Two similar spheres having +Q and -Q charges are kept at a certain distance. F force acts between the two. If at the middle of two spheres, another similar sphere having +Q charge is kept, then it experiences a force in magnitude and direction as ______.
Five charges, q each are placed at the corners of a regular pentagon of side ‘a’ (Figure).
(a) (i) What will be the electric field at O, the centre of the pentagon?
(ii) What will be the electric field at O if the charge from one of the corners (say A) is removed?
(iii) What will be the electric field at O if the charge q at A is replaced by –q?
(b) How would your answer to (a) be affected if pentagon is replaced by n-sided regular polygon with charge q at each of its corners?
