A rectangular loop with dimensions 4.90 cm by 17.0 cm is placed in a uniform electric field and rotated to find the orientation that produces a maximum electric flux through the loop of 7.73 ✕ 104 N · m2/C. What is the magnitude of this electric field?
Why did the rectangle join the circus? It wanted to find its electric field orientation and become a "current" performer!
To find the magnitude of the electric field, let's use a clown trick called Gauss' law. Gauss' law states that the electric flux through a closed surface is directly proportional to the charge enclosed by that surface.
In this case, the electric flux through the rectangular loop can be given as:
Electric Flux = Electric Field * Area
We can rearrange this equation to solve for the electric field:
Electric Field = Electric Flux / Area
Given that the electric flux is 7.73 ✕ 10^4 N · m^2/C, and the area of the rectangular loop is (4.90 cm * 17.0 cm), we can plug in the values and calculate the electric field:
Electric Field = (7.73 ✕ 10^4 N · m^2/C) / (4.90 cm * 17.0 cm)
Conversion time! Since meters and centimeters don't like to play together, let's convert the measurements to the same unit:
Electric Field = (7.73 ✕ 10^4 N · m^2/C) / (0.049 m * 0.17 m)
Now let's do the math:
Electric Field = 2.00 ✕ 10^6 N/C
So the magnitude of the electric field in this situation is 2.00 ✕ 10^6 N/C. That's quite shocking, isn't it?
To find the magnitude of the electric field, we need to use the equation for the electric flux through a loop. The formula for electric flux is given by:
Phi = E * A * cos(theta)
Where:
- Phi is the electric flux,
- E is the magnitude of the electric field,
- A is the area of the loop, and
- theta is the angle between the electric field and the normal to the loop.
Given:
- Electric flux (Phi) = 7.73 * 10^4 N · m2/C
- Dimensions of the loop: length (L) = 17.0 cm and width (W) = 4.90 cm
We are trying to find the magnitude of the electric field (E).
To find the area of the loop (A), we can multiply its length (L) by its width (W):
A = L * W
Next, we need to find the value of cos(theta) when the loop is in the orientation that produces the maximum electric flux. The maximum flux occurs when the electric field is parallel to the normal of the loop, so cos(theta) = 1.
Now, we can rearrange the electric flux equation to solve for the electric field (E):
E = Phi / (A * cos(theta))
Substituting the values we have:
E = (7.73 * 10^4 N · m2/C) / [(17.0 cm) * (4.90 cm) * 1]
Now, let's convert the length and width from centimeters to meters:
E = (7.73 * 10^4 N · m2/C) / [(0.17 m) * (0.049 m)]
Finally, we can calculate the magnitude of the electric field:
E ≈ 2.823 * 10^6 N/C
Therefore, the magnitude of the electric field is approximately 2.823 * 10^6 N/C.