Question

State Coulomb's law in vector form.

Answer 1

Coulomb’s law: The electrostatic force between two stationary point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Vector form of Coulomb’s law: if ri and rj are position vectors of two point charges Qi and Qj and Fij is the force exerted by Qj on Qi, then

Fij=kQiQj(ri-rj)/[|ri-rj|]^3.

Importance of vector form: (1):Vector form gives the direction of the force along with magnitude.

(2) When we study force on a given charge by many charges, we use superposition principle. This principle states that electrostatic force on a given charge due to many charges is vector sum of individual forces due to other charges. When we consider effect due to one charge we ignore the presence of other charges. The force on charge 1 due to charges 2,3,…..N is ,according to superposition principle, is given by

F=F12+F13+……….F1N.

Answer 2

The Coulomb’s law written as

F=kq1q2r2

gives the magnitude of the static electric force between the two charges, exerted on each charge. Here r is the shortest distance between the charges.

The direction of force on any of the charges is pointed along the line joining the positions of the two charges. So, let’s call the position vectors of the two charges r1→ and r2→, the difference between the two positions is the vector r2→−r1→, which points from q1 to q2. We’ll call this r12→ and appropriately we’ll call the reverse of this vector r21→. Both of these vectors point along the line joining the two charges.

But we will only require the unit vectors, so that we can simply scale the unit vector by the magnitude of the force. So we have

r12^=r12→r where r=r12 as the magnitude of the r12 is nothing but the shortest distance between the two positions. Thus we can now write the force vectors:

F12→=kq1q2r2⋅r21^ (Force on q1 by q2)

and

F21→=kq1q2r2⋅r12^ (Force on q2 by q1)