Principle of superposition|Electric field

5. Principle Of Superposition

  • Coulomb's law gives the electric force acting between two electric charges.

  • Principle of superposition gives the method to find force on a charge when system consists of large number of charges.

  • According to this principle when a number of charges are interacting the total force on a given charge is vector sum of forces exerted on it by all other charges.

  • This principle makes use of the fact that the forces with which two charges attract or repel one another are not affected by the presence of other charges.

  • If a system of charges has n number of charges say q1, q2, ...................., qn, then total force on charge q1 according to principle of superposition is
    F = F12 + F13 + .................................. F1n
    Where F12 is force on q1 due to q2 and F13 is force on q1 due to q3 and so on.

    Force due to system of multiple charges

  • F12, F13, .................. F1n can be calculated from Coulomb's law i. e.
    $$\boldsymbol{F_{12}}=\frac{kq_{1}q_{2}\boldsymbol{\widehat{r}_{12}}}{4\pi \epsilon _{0}r_{12}^{2}}$$
    $$\boldsymbol{F_{1n}}=\frac{kq_{1}q_{2}\boldsymbol{\widehat{r}_{1n}}}{4\pi \epsilon _{0}r_{1n}^{2}}$$
  • The total force F1 on the charge q1 due to all other charges is the vector sum of the forces F12, F13, ................................. F1n.
    F1 = F12 + F13 + ..................................
    Principle of superposition|Electric field
  • The vector sum is obtained by parellogram law of addition of vector.

  • Similarly force on any other charge due to remaining charges say on q2, q3 etc. can be found by adopting this method.

6. Electric Field

  • Electrical interaction between charged particles can be reformulated using the concept of electric field.

  • To understand the concept consider the mutual repulsion of two positive charged bodies as shown in fig (a)

    Principle of superposition|Electric field

  • Now if remove the body B and label its position as point P as shown in fig (b), the charged body A is said to produce an electric field at that point (and at all other points in its vicinity)

  • When a body B is placed at point P and experiences force F, we explain it by a point of view that force is exerted on B by the field not by body A itself.

  • The body A sets up an electric field and the force on body B is exerted by the field due to A.

  • An electric field is said to exists at a point if a force of electric origin is exerted on a stationary charged (test charge) placed at that point.

  • If F is the force acting on test charge q placed at a point in an electric field then electric field at that point is
    E = F/q
    or F = qE

  • Electric field is a vector quantity and since F = qE the direction of E is the direction of F.

  • Unit of electric field is (N.C-1)

Q. Find the dimensions of electric field
Ans. [MLT-3A-1]

7. Calculation of Electric Field

  • In previous section we studied a method of measuring electric field in which we place a small test charge at the point, measure a force on it and take the ratio of force to the test charge.

  • Electric field at any point can be calculated using Coulomb's law if both magnitude and positions of all charges contributing to the field are known.

  • To find the magnitude of electric field at a point P, at a distance r from the point charge q, we imagine a test charge q'to be placed at P. Now we find force on charge q' due to q through Coulomb's law.
    $$\boldsymbol{F}=\frac{kqq_{'}}{4\pi \epsilon _{0}r^{2}} $$
    electric field at P is $$\boldsymbol{E}=\frac{kqq_{'}}{4\pi \epsilon _{0}r^{2}} $$
    The direction of the field is away from the charge q if it is positive

    Principle of superposition|Electric field

  • Electric field for either a positive or negative charge in terms of unit vector r directed along line from charge q to point P is $$\boldsymbol{F}=\frac{kqq_{'}\boldsymbol{\widehat{r}}}{4\pi \epsilon _{0}r^{2}} $$
    r = distance from charge q to point P.

  • When q is negative , direction of E is towards q, opposite to r.

    Electric Field Due To Multiple Charges

  • Consider the number of point charges q1, q2,........... which are at distance r1P, r2P,................... from point P as shown in fig

    resultant electric field due to two charges at a point P

  • The resultant electric field is the vector sum of individual electric fields as
    E = E1P + E2P + .....................
    Principle of superposition|Electric field

    This is also a direct result of principle of superposition discussed earlier in case of electric force on a single charge due to system of multiple charges.

  • E is a vector quantity that varies from one point in space to another point and is determined from the position of square charges.

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