Calculating the Components of the Net Electric Field at Point P

In this article, we will walk through the process of determining the coordinates of the net electric field at a specific point in space. Specifically, we will focus on point P, located at y 6.00 m, and explore the contributions of multiple charges to the electric field at that point.

Introduction

In order to calculate the (x)- and (y)-components of the net electric field at point P, which is positioned at (y 6.00 m), we need to know several key details:

The positions and magnitudes of the charges producing the electric field. The distances from these charges to point P. The direction of the electric fields produced by each charge at point P.

By understanding these prerequisites, we can use Coulomb's Law to determine the electric field components and apply principles of linear superposition to find the net electric field at point P.

Analysis and Calculation

Consider the configuration of charges arranged in such a way that their electric fields exhibit symmetry with respect to the y-axis. Given that the charges are of equal magnitude but opposite signs and are located at the same vertical distance from point P, we can use the principle of linear superposition to simplify our calculations.

Step 1: Determine the y-components of the electric fields.

Since the charges are symmetrical about the y-axis, the y-components of their electric fields will cancel each other out at point P. This is a direct result of the charges being equal in magnitude but opposite in sign.

Step 2: Calculate the x-components of the electric fields.

For one of the charges, denote the distance from the charge to point P as (r). Using Coulomb's Law, the magnitude of the electric field at point P due to a single charge is given by:

(E_1 frac{1}{4piepsilon_0}frac{q}{r^2})

where (epsilon_0) is the permittivity of the medium (in vacuum, (3times10^{-11} text{F/m})), and (r sqrt{(2.5 text{ m})^2 (6.00 text{ m})^2} 6.5 text{ m}).

Substituting the values:

(E_1 frac{1}{4pitimes3times10^{-11} text{F/m}}timesfrac{1.6 times 10^{-19} text{C}}{(6.5 text{ m})^2})

Simplifying further:

(E_1 frac{1}{3.14 times 3 times 4 times 10^{-11} text{F/m}}timesfrac{1.6 times 10^{-19} text{C}}{42.25 text{ m}^2}) ≈ (1.309 times 10^{-11} text{ N/C})

The (x)-component of this vector field from one charge is:

(E_{1x} -E_1 cdot frac{2.5}{r})

Substituting the values:

(E_{1x} -1.309 times 10^{-11} text{ N/C} cdot frac{2.5 text{ m}}{6.5 text{ m}^3}) ≈ (-2.618 times 10^{-11} text{ N/C})

Since the charges are symmetrical about the y-axis, the resultant (x)-component is twice the x-component of one charge:

(E_x 2 cdot E_{1x}) ≈ (-5.236 times 10^{-11} text{ N/C})

Conclusion

By using the principles of linear superposition and Coulomb's Law, we have calculated the net electric field's components at point P. The calculations show that the y-components cancel out due to symmetry, and the x-component is determined by the configuration and distances of the charges.

Understanding these calculations is crucial for anyone working in the fields of physics and engineering, especially when dealing with complex charge distributions and electric fields.

Related Keywords

electric field Coulomb's Law permittivity of vacuum linear superposition

References

1. Griffiths, D.J. (2017) Introduction to Electrodynamics, 4th Ed. Cambridge University Press. 2. Halliday, D., Resnick, R., Walker, J. (2014) Fundamentals of Physics, 10th Ed. Wiley.