A point particle with charge q=4.5 μC is placed on the x...

A point particle with charge $q=4.5 \mu \mathrm{C}$ is placed on the $x$ axis at $x=-10 \mathrm{cm}$ and a second particle of charge $q=6.7 \mu \mathrm{C}$ is placed on the $x$ axis at $x=+25 \mathrm{cm} .$ a. Determine the $x$ and $y$ components of the electric field due to this arrangement of charges at the point $(x, y)=(10,10)$ (the units here are centimeters). b. Determine the magnitude and direction of the electric field at this point.

A point particle with charge $q=4.5 \mu \mathrm{C}$ is placed on the $x$ axis at $x=-10 \mathrm{cm}$ and a second particle of charge $q=6.7 \mu \mathrm{C}$ is placed on the $x$ axis at $x=+25 \mathrm{cm} .$
a. Determine the $x$ and $y$ components of the electric field due to this arrangement of charges at the point $(x, y)=(10,10)$ (the units here are centimeters).
b. Determine the magnitude and direction of the electric field at this point.

Key Concepts

Magnitude and Direction Calculation

After obtaining the vector components of the electric field, the overall magnitude can be found using the Pythagorean theorem, and its direction can be determined by calculating the angle using inverse trigonometric functions. This conversion from components to magnitude and direction is essential for fully characterizing the vector field.

Vector Decomposition

Since the electric field is a vector quantity, it is often necessary to break it down into its components (e.g., x and y components in a Cartesian coordinate system). This involves using trigonometric relationships to resolve vectors into orthogonal directions, which simplifies the process of combining fields from different sources.

Electric Field

The electric field is a vector quantity that represents the force per unit charge exerted on a positive test charge placed within the field. It has both magnitude and direction, making it necessary to consider its vector nature when multiple charges are involved.

Superposition Principle

The superposition principle in electrostatics states that the total electric field created by a system of charges is the vector sum of the electric fields produced by each individual charge. This allows for the separate calculation of fields from multiple charges, which can then be added together to find the overall field at a point.

Coulomb's Law

Coulomb's Law provides the fundamental relationship to calculate the electric field produced by a point charge. It states that the electric field (or force) is directly proportional to the magnitude of the charge and inversely proportional to the square of the distance from the charge, with the proportionality constant being the Coulomb constant (k). This law is the basis for calculating the contributions of each point charge to the overall field.

Transcript

00:01 For this problem, we can plot the graph to look at this setup.

00:07 In the right graph, we can see there is q1 and q2 on each side.

00:14 And we want to calculate the electric field at a given point 1010.

00:20 So for the electric field, we can calculate the electric field from each part separately.

00:28 So q1 giving the electric field e1 and q2 gives the electric field e2.

00:37 So the total electric field is e1 plus e2.

00:43 So we can write the equation for the q1 and q2, which is q1 is equal to q1 divided by the 4 pi epsilon 0 and times r1 square r1 is a distance from q1 to given point 1010 and if we write this into the vector equation so this also equals the r1 hat for the direction and e2 is just the same as e1 just the difference is that is q2 divide by four point xcelon r i2 square times i2 hat for the direction.

01:42 So we now each part, so we can easily calculate this.

01:49 But it's more easier to calculate the component for each electric field.

01:57 For example, e1x is the electric field e1 in the x component.

02:04 So which is just equal to e1 rector times the x component.

02:15 And we can solve this equation, use the number, and which is equal to 7, 2, 3, 4, 8, 4, newton.

02:34 That is for the e1, x component, and the e1, y component is same as that.

02:43 So it is e1 vector times y component, and which is easily calculated given this number...