A circular surface with a radius of 0.058 m is exposed to a uniform external electric field of magnitude 1.49 104 N/C. The electric flux through the surface is 74 N · m2/C. What is the angle between the direction of the electric field and the normal to the surface?
To find the angle between the direction of the electric field and the normal to the surface, we can use the formula for electric flux:
Electric flux = Electric field * Surface area * Cosine(theta)
Where:
Electric flux is given as 74 N · m^2/C
Electric field is given as 1.49 * 10^4 N/C
Surface area can be calculated using the formula for the area of a circle: pi * radius^2
Theta is the angle between the electric field and the normal to the surface (what we're trying to find)
Let's substitute the given values into the formula and solve for theta:
74 N · m^2/C = (1.49 * 10^4 N/C) * (pi * (0.058 m)^2) * Cosine(theta)
First, let's calculate the surface area:
Surface area = pi * radius^2 = pi * (0.058 m)^2
Surface area ≈ 0.0106 m^2
Now let's rearrange the equation to solve for cosine(theta):
Cosine(theta) = (74 N · m^2/C) / ((1.49 * 10^4 N/C) * (pi * (0.058 m)^2))
Cosine(theta) ≈ 0.00737
To find the value of theta, we need to take the arccosine (inverse cosine) of 0.00737:
Theta ≈ arccos(0.00737)
Using a calculator, the inverse cosine of 0.00737 is approximately 89.736 degrees.
Therefore, the angle between the direction of the electric field and the normal to the surface is approximately 89.736 degrees.
To find the angle between the direction of the electric field and the normal to the surface, we can use the formula for electric flux:
Electric Flux = Electric Field x Surface Area x Cos(θ),
where θ is the angle between the electric field vector and the normal vector.
Given:
Radius (r) = 0.058 m,
Electric Field (E) = 1.49 * 10^4 N/C,
Electric Flux (Φ) = 74 N · m^2/C.
We can rearrange the formula to solve for the angle θ:
Cos(θ) = Electric Flux / (Electric Field x Surface Area).
Now, let's calculate surface area and solve for the angle:
Surface Area = π * r^2,
Surface Area = π * (0.058 m)^2.
Surface Area ≈ 0.0106 m^2.
Plugging the values into the formula:
Cos(θ) = 74 N · m^2/C / (1.49 * 10^4 N/C * 0.0106 m^2).
Cos(θ) ≈ 0.4434.
To find the angle θ, we take the inverse cosine (arccos) of 0.4434:
θ = arccos(0.4434).
Using a calculator or trigonometric table, we find:
θ ≈ 63.62 degrees.
Therefore, the angle between the direction of the electric field and the normal to the surface is approximately 63.62 degrees.