A flat 1.0m^2 surface is vertical at x=2.0m and parallel to the yz-plane. What is the flux through the surface if it is located in a uniform electric field given by E= 26.0i + 42.0j + 62.0k N/C?
Im having trouble understanding the question, I know how to do vectors but this question is confusing me. Any help would be fantastic!
To find the flux through the given surface in the uniform electric field, we can use the formula:
Flux (Φ) = ∫∫ E⋅dA
Where Φ is the flux, E is the electric field vector, and dA is the vector representing the infinitesimal area element.
Here's a step-by-step breakdown of how to solve the problem:
Step 1: Understand the given information.
The surface given is a flat 1.0 m² surfacethat is vertical at x = 2.0 m and parallel to the yz-plane. The electric field is given by E = 26.0i + 42.0j + 62.0k N/C.
Step 2: Find the normal vector to the surface.
Since the surface is vertical and parallel to the yz-plane, the normal vector to the surface is along the x-axis. Therefore, the normal vector can be written as n = i.
Step 3: Calculate the flux.
To calculate the flux, we need to integrate the dot product of the electric field and the infinitesimal area vector over the surface. The infinitesimal area vector (dA) is equal to the magnitude of the surface area (1.0 m²) multiplied by the normal vector (n), so dA = 1.0 * i.
The dot product of the electric field (E) and the infinitesimal area vector (dA) is:
E⋅dA = (26.0i + 42.0j + 62.0k)⋅(1.0i) = 26.0
Since the dot product is constant and does not depend on the position on the surface, the flux through the surface is given by:
Φ = E⋅dA = 26.0
Therefore, the flux through the given surface in the uniform electric field is 26.0 N∙m²/C.
Remember to include the appropriate units in your answer.
Sure! Let's break down the problem step by step.
First, let's understand what is meant by flux. Flux is a measure of how much electric field passes through a given surface. Mathematically, flux can be calculated as the dot product of the electric field vector and the surface area vector.
In this problem, we are given a flat surface that is vertical at x=2.0m and parallel to the yz-plane. The surface area is given as 1.0m^2. We are also given the electric field vector, E = 26.0i + 42.0j + 62.0k N/C.
To calculate the flux, we need to determine the component of the electric field vector that is perpendicular to the surface. Since the surface is parallel to the yz-plane, we only need to consider the x-component of the electric field, which is 26.0i N/C.
Now, we can calculate the flux using the formula: Flux = E dot A, where E is the electric field vector and A is the surface area vector.
The surface area vector, A, is a vector that points normal (or perpendicular) to the surface. In this case, since the surface is vertical, the surface area vector will be in the -x direction, which is -i.
Now, let's calculate the flux:
Flux = E dot A
= (26.0i) dot (-i)
= -26.0
So, the flux through the surface is -26.0 Nm^2/C.
I hope this explanation helps! Let me know if you have any further questions.