Two positive point charges q are placed on the x-axis, one...

Two positive point charges $q$ are placed on the $x$-axis, one at $x = a$ and one at $x = -a$. (a) Find the magnitude and direction of the electric field at $x = 0$. (b) Derive an expression for the electric field at points on the x-axis. Use your result to graph the $x$-component of the electric field as a function of $x$, for values of $x$ between $-$4a and $+$4a.

Key Concepts

Graphical Analysis of Electric Fields

Graphing the electric field as a function of position helps in visually understanding how the field varies with distance and direction from the source charges. It provides insights into the regions of influence, zero-field points, and areas where the electric field undergoes significant changes, which is critical in both teaching and practical applications.

Superposition Principle

The superposition principle states that the total electric field created by multiple charges is equal to the vector sum of the fields produced by each individual charge. This is a crucial concept when analyzing systems with more than one charge, as it allows for the computation of the net field by simply adding the individual fields, taking into account their magnitudes and directions.

Symmetry in Electrostatics

Symmetry considerations play a significant role in simplifying the analysis of electric fields. In configurations where charges are symmetrically placed, such as on the x-axis at equidistant points, symmetry can be used to predict directions of field components and to cancel out certain vector contributions, thereby simplifying the mathematical calculations.

Coulomb's Law

This fundamental law describes the electric force between two point charges. It expresses that the magnitude of the force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them. The concept is extended to define the electric field created by a point charge, indicating how that field diminishes with distance.

Electric Field of a Point Charge

An electric field represents the force per unit charge at any point in space due to a source charge. The electric field generated by a point charge is given by the equation E = kq/r², where r is the distance from the charge. This concept allows for the determination of the influence of a single charge on its surroundings.

Transcript

00:01 Okay, so for the first part, let's say this is our x -axis, this is positive x -axis, this is negative x -axis, and this will be our y -axis.

00:15 So this is the positive y -axis.

00:17 And the two charges are at x -equal to plus a and x -equal to minus a.

00:23 Let's call the charges to be 1 and 2, but the magnitudes of both the charges are equal and is equal to q.

00:32 Now we need to find the electric field here at the center where x is equal to 0.

00:43 So let's call this point to be p.

00:48 So electric field due to charge at 1 will be equal to kq square over a square and this electric field will be would point in the positive x -axis hence plus i cap.

01:09 Similarly, electric field at point p due to charge at 2 will be equal to k sq square over a square because they are placed at the same distance a from point p and they have the same magnitudes and the magnitude hence the magnitudes of the electric field due to charge at 1 and 2 will be equal.

01:31 However, the direction of this electric field due to charge at 2 will be along the negative x direction.

01:41 Hence, here we will have negative icap.

01:47 Now, using superposition principle, the net electric field at p will be e1 plus e2 and that turns out to be equal to 0.

01:56 Hence there is no electric field at the midpoint of the two charges.

02:04 Now for the second part, so let me just draw both the axis first.

02:15 So the charges are at a and minus a, let's call this one, charge one, charge two, and we need to find the electric field at a distance x from the origin.

02:29 Now if x mod is less than a, which means that the electric field, we need to find the electric field at the point that is distanced in between these two places.

02:47 It should be either here or here, but it should be, the magnitude should be less than a.

02:53 So let's say that the point p where we need to find the electrical field is somewhere here...

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