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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 q_{1}, q_{2}, ...................., q_{n}, then total force on charge q_{1} according to principle of superposition is
F = F_{12} + F_{13} + .................................. F_{1n}
Where F_{12} is force on q_{1} due to q_{2} and F_{13} is force on q_{1} due to q_{3} and so on.
- F_{12}, F_{13}, .................. F_{1n} can be calculated from Coulomb's law i. e.
$$\boldsymbol{F_{12}}=\frac{kq_{1}q_{2}\boldsymbol{\widehat{r}_{12}}}{4\pi \epsilon _{0}r_{12}^{2}}$$
to,
$$\boldsymbol{F_{1n}}=\frac{kq_{1}q_{2}\boldsymbol{\widehat{r}_{1n}}}{4\pi \epsilon _{0}r_{1n}^{2}}$$
- The total force F_{1} on the charge q_{1} due to all other charges is the vector sum of the forces F_{12}, F_{13}, ................................. F_{1n}.
F_{1} = F_{12} + F_{13} + ..................................
- The vector sum is obtained by parellogram law of addition of vector.
- Similarly force on any other charge due to remaining charges say on q_{2}, q_{3} 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)
- 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
^{-3}A
^{-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
- 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 q_{1}, q_{2},........... which are at distance r_{1P}, r_{2P},................... from point P as shown in fig
- The resultant electric field is the vector sum of individual electric fields as
E = E_{1P} + E_{2P} + .....................
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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