E2 (total) of rod (of length L = 9.15 mm) = 4.18 ×10^3 V/m. E1=1.76 x 10^3. E3=1.57 x 10^3. The rod consists of three sections of the same material but with different radii. The radius of section 3 is 2.27 mm. What is the radius of (a) section 1 and (b) section 2?
If someone could please help???
To solve this problem, we can start by using the formula for electric field (E) inside a uniformly charged conductor:
E = (k * Q) / (r^2)
where:
- E is the electric field
- k is the Coulomb's constant (k = 9 x 10^9 N m^2/C^2)
- Q is the total charge enclosed by the Gaussian surface
- r is the distance from the center of the conductor
Since the rod consists of three sections of the same material, the value of k is the same for each section. We can set up the following equations using the given information:
E1 = (k * Q1) / (r1^2)
E2 = (k * Q2) / (r2^2)
E3 = (k * Q3) / (r3^2)
We are given the values of E1, E2, and E3, and we can figure out the values of Q1, Q2, and Q3 by multiplying the electric field by the corresponding area element.
Q1 = E1 * (Area1) = E1 * (π * r1^2)
Q2 = E2 * (Area2) = E2 * (π * r2^2)
Q3 = E3 * (Area3) = E3 * (π * r3^2)
Now, we can substitute the values of Q1, Q2, and Q3 in the equations:
E1 = (k * E1 * (π * r1^2)) / (r1^2)
E2 = (k * E2 * (π * r2^2)) / (r2^2)
E3 = (k * E3 * (π * r3^2)) / (r3^2)
Simplifying these equations:
1 = k * (π * r1^2)
1 = k * (π * r2^2)
4.18 × 10^3 = k * (π * r3^2)
We can solve these equations to find the values of r1 and r2:
r1 = √(1 / (k * π))
r2 = √(1 / (k * π))
Now, we need the value of k, which is Coulomb's constant. We can substitute the value of k into the equations to find the values of r1 and r2.
k = 9 x 10^9 N m^2/C^2
Substituting this value into the equations:
r1 = √(1 / (9 x 10^9 * π))
r2 = √(1 / (9 x 10^9 * π))
Calculating these values, we can find the radius of section 1 and section 2.