All About Derivation Of Electric Field Due To A Point Charge

The intensity of an electric field at a point due to a group of charges is equal to the arithmetic sum of intensities of an electric field due to each charge at the same given time. An electric field is a space around the electric charges where the force of either attraction or repulsion can be felt. Whereas an electric charge is the property of matter which governs how the matter will react in the presence of an electric field. When an electric charge (say q_o) is held in charge (say Q), it will experience a force of attraction or repulsion. Let’s study the equation for the electric field of a point charge and its derivation. 

Coulomb’s Law and Electric Field

The laws of Coulomb deal with the force that acts between two electric charges. This law is only applicable for a static (imbalanced) charge. The electric field is the production of motion of these charges, and it propagates through space at the speed of light. According to the Coulomb’s law:

• The electric charges will modify its surroundings by forming an electric field around it.
• If we introduce a new charge in this region, that charge will experience some force due to the electric field produced by the previous charges. 

The Formula for Force of Electric Field due to Point Charge

Suppose an electric charge qo is present in the vicinity of charge Q. It will either experience an attraction or repulsion within. This force shall be due to the electric field built around the charge Q. Thus, the formula for it will be as follows:

F= kQq_1/r²

r = The distance of separation between source charge and test charge

Q = source charge,

q₁ = test charge, and

k= constant (9×10⁹Nm²C⁻¹)

Example of Electric Field (Due to a Point Charge)

Assume a point charge as ‘Q’ present at the origin O. This charge is placed in a vacuum. According to Coulomb’s law, if we place another point charge (say q) at a position (say P), if OP=r, then a force will be exerted by Q on the q charge. Thus, Q will produce an electric field around its surroundings, and this new charge q on point P will generate a force. 

Solved Questions on Electric Field due to Point Charge

Q 1- State the magnitude of a point charge in an electric field if the point charges of 30μC were at a distance of 2 m.

Solution:

The formula for the calculation of the magnitude of an electric field due to a point charge will be as follows:

E= k q/d²

= (9×109) (30×10⁻⁶)/(2)²

=1.08× 10⁶ N/C. 

Answer: The magnitude of a point charge in an electric field, if the point charges of 30μC were at a distance of 2 m, will be 1.08× 106 N/C. 

Q 2- Calculate the magnitude of an electric field at a point (in the middle of two charges). These 2 point charges are 5μC and -3.2μC, separated by a distance of 6cm. 

Solution:

Assume the line that joins the charges is the x-axis. Therefore, we will find the electric field due to the charge at the midpoint (d=2cm). 

The magnitude of the electric field will be as follows:

E= k q/d²

= (9×109) (5×10⁻⁶)/(o.o3)²

=4.05×10 N/C. 

Answer: The magnitude of an electric field at a point (in the middle of two charges) will be 4.05×10 N/C. 

Q 3- Calculate both the magnitude and direction of an electric field at 4cm. It is left to the point charge of -4.4 nC.

Solution:

There is no need for a sign while calculating the magnitude, as we have to take the absolute values only. Therefore, we will ignore the minus sign. 

E= k q/d²

E= (9×10⁹) (4.4×10^-9)/(o.o4)²

E= 6.336× 10^-2 N/C.

The negative charge represents the direction of the electric field towards the charge. Thus, we can determine the direction by looking at the charge signs.

Conclusion

The electric field is the surrounding space of charges in which a force of attraction or repulsion will be felt due to the force exerted by these charges. Coulomb’s laws deal with the forces acting between two electric charges. According to it, if we introduce a new charge in this region, that charge will experience some force due to the electric field produced by the previous charges. We can determine or calculate the magnitude of the charge of an electric field by the formula E= k q/d².