A charge q 1=+2 q^q1=+2 q is at the origin, and a charge q...

A charge $q 1=+2 q^{q_{1}}=+2 q$ is at the origin, and a charge $q 2=-q^{q_{2}}=-q$ is on the $x$ axis at $x=a$. Find expressions for the total electric field on the $x$ axis in each of the regions (a) $x<0 x<0$ (b) $0<x<a 0<x<a$ and (c) $a<x a<x$ (d) Determine all points on the $x$ axis where the electric field is zero. (e) Use a graphing calculator or spreadsheet to make a plot of $E_{x}$ versus $x$ for all points on the $x$ axis, and (f) qualitatively discuss what happens for $-\infty<x<\infty$. $-\infty<x<\infty$. Example 16-6

A charge $q 1=+2 q^{q_{1}}=+2 q$ is at the origin, and a charge $q 2=-q^{q_{2}}=-q$ is on the $x$ axis at $x=a$. Find expressions for the total electric field on the $x$ axis in each of the regions (a) $x<0 x<0$ (b) $0<x<a 0<x<a$ and (c) $a<x a<x$
(d) Determine all points on the $x$ axis where the electric field is zero. (e) Use a graphing calculator or spreadsheet to make a plot of $E_{x}$ versus $x$ for all points on the $x$ axis, and (f) qualitatively discuss what happens for $-\infty<x<\infty$. $-\infty<x<\infty$. Example 16-6

Key Concepts

Equilibrium Points (Zero Fields)

Finding the points where the electric field is zero involves setting the net field (the vector sum of individual fields) equal to zero and solving for the position(s). These equilibrium points are areas where the forces due to individual charges cancel each other out, which is crucial when studying the stability of a charge in an electric field or analyzing motion in electrostatic configurations.

Piecewise Analysis of Electric Fields

Piecewise analysis involves dividing the space into regions based on the positions of the charges and analyzing the behavior of the electric field in each region separately. This method is used when the expressions for the electric field vary depending on the relative positions of the observation point and the charges, enabling a systematic approach to solving field equations in different spatial intervals.

Qualitative Graphical Analysis

Qualitative graphical analysis involves plotting the electric field as a function of position and interpreting the behavior of the plot to gain insights into how the electric field changes across different regions. This approach is important for visualizing phenomena such as field reversals, regions of strong and weak fields, and asymptotic behavior at large distances, which collectively aid in understanding the overall physical system.

Electric Field from a Point Charge

The electric field produced by a point charge is a vector quantity that points away from positive charges and toward negative charges. It can be calculated using the equation E = k|q|/r^2, where k is Coulomb's constant, q is the charge, and r is the distance from the charge. This concept is essential for understanding how individual charges contribute to the total electric field in space.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges, stating that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. This law is fundamental for calculating the magnitude and direction of the electric field generated by a point charge, which is critical for solving problems involving electric fields from discrete charges.

Superposition Principle

The principle of superposition states that the total electric field created by multiple charges is the vector sum of the fields produced by each individual charge. This is a key concept in electrostatics, allowing one to analyze complex configurations by considering the contribution from each charge separately and then combining the results to find the net field.

Transcript

00:01 For this problem on the topic of electrostatics, we have a charge q1, which is plus 2q at the origin, and a charge q2, which is minus q on the x -axis at x is equal to a.

00:11 And we want to find the expressions for the total electric field on the x -axis in each of the regions, x less than 0, x between 0 and a and x greater than a, and then determine all the points on the x -axis where the electric field is 0.

00:23 We then want to make a plot of e versus x for all points on the x axis and qualitatively discuss what happens between minus infinity and infinity.

00:36 Now, firstly, for part a, we want the x component of the field x, and this is equal to the x component of the field due to 1 plus the x component of the field due to charge 2.

00:52 And this is equal to minus k -k -1 over x squared plus k -2 over a plus x all squared.

01:09 And we can write this as k -into minus q -1 over x squared plus the magnitude of charge 2, q2 over the distance to the point a plus x all squared.

01:29 And so we can then write this as k into minus 2q over x squared plus q over a plus x all squared.

01:47 And so we have an expression for the electric field in the first region.

01:55 In the next region, we have the electric field x is equal to, again, e1x plus e2x, which in this case is k times q1 over x squared plus k times q2 over a minus x or squared.

02:22 This is equal to k into the magnitude of q1 divided by x squared plus the magnitude of q2 divided by a minus x all squared.

02:38 Putting in the value for the charges, this is k into 2k over x squared plus q divided by a minus x all squared.

02:57 And finally, for the third region, for part c, we again have the net field, x, to be the contribution due to charge 1 plus the contribution due to charge 2...

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