A charge of q1=1×10−6 C is located at corner A of an equilateral triangle. A charge q2=4×10−6 C is located at corner B and a charge of q3=7×10−6 C is located at corner C. The distance AB=0.46 m (the other two sides of the triangle are the same). See the figure.
a) What is the x-component of the Electric field (in V/m) at point P which is located half way between A and C?
b)What is the y-component of the Electric field (in V/m) at point P?
c)What is the Electric Potential (in Volts) at point P?
d)What is the Electric Potential Energy (configuration energy) in Joules of this system of 3 charges?
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E =kq/r²
(a) E(x) = E₁-E₃=kq₁/r² - kq₃/r² =
=9•10⁹{(1•10⁻⁶/0.23²)- (7•10⁻⁶/0.23²)}= -1.02•10⁶ V/m
(b) An altitude of a triangle is
h=0.46cos30=0.46•0.866=0.398 m
E(y) = kq ₂/h²= 9•10⁹•4•10⁻⁶/0.398²= - 2.27•10⁵ V/m
(c)
φ=φ₁+ φ₂+ φ₃= kq₁/r + kq₃/h+ kq₃/r=
=9•10⁹{ (1•10⁻⁶/0.23 )+ (4•10⁻⁶/0.398) + (7•10⁻⁶/0.23)} = …
(d)
PE=PE₁₂+PE₁₃+PE₂₃=
= kq₁q₂/r +kq₁q₃/r + kq₂q₃/r=…
To solve this problem, we can use the concept of Coulomb's law to calculate the electric field at point P due to each of the charges.
a) The x-component of the Electric field at point P can be calculated by considering the Coulomb's law between points A and P (EPa), and between points C and P (EPc).
The formula for the electric field due to a point charge, q, at a distance r is given by:
E = k*q / r^2
where k is the electrostatic constant, which is approximately 8.99 x 10^9 N m^2 / C^2.
The x-component of the Electric field at point P due to q1 can be calculated as:
EPa = (k * q1) / (AP^2), where AP is the length of the side of the equilateral triangle.
The x-component of the Electric field at point P due to q2 can be calculated as:
EPb = (k * q2) / (BP^2), where BP is the length of the side of the equilateral triangle.
The x-component of the Electric field at point P due to q3 can be calculated as:
EPc = (k * q3) / (CP^2), where CP is the length of the side of the equilateral triangle.
Since point P is located halfway between points A and C, the distance AP and CP are equal to AP = CP = (0.5) * (side of the triangle). We can use this information to calculate the x-component of the electric field at point P.
Using the given values:
q1 = 1x10^(-6) C,
q2 = 4x10^(-6) C,
q3 = 7x10^(-6) C,
AP = CP = (0.5) * side of triangle = (0.5) * 0.46 m,
we can substitute the values into the equations for calculating the x-component of the electric field at point P due to each charge.
b) The y-component of the Electric field at point P can be calculated similarly to the x-component by using the same equations, except we consider the y-direction instead.
c) The Electric Potential at point P can be calculated by summing up the potential due to each charge at point P. The potential due to each charge q at a distance r is given by:
V = k*q / r
Using the given values of q1, q2, q3, and the distance between each charge and point P (0.5 * side of the triangle), we can calculate the Electric Potential at point P.
d) The Electric Potential Energy of the system can be calculated by summing up the contribution of each pair of charges. The potential energy between two charges q1 and q2 is given by:
U = k * q1 * q2 / r12
Similarly, we can calculate the potential energy between q1 and q3, and q2 and q3 using the given values.
Let's calculate these values step by step:
Step 1: Calculate the side length of the equilateral triangle, using the distance AB.
Step 2: Calculate the x-component and y-component of the Electric field at point P.
Step 3: Calculate the Electric Potential at point P.
Step 4: Calculate the Electric Potential Energy of the system.
To find the components of the electric field at point P, we need to consider the electric field contributions from each charge individually and then combine them.
a) The x-component of the electric field at point P:
1. Start by finding the x-component of the electric field due to charge q1 at point P. The electric field due to a point charge at a distance r is given by the formula E = k * q / r^2, where k is the electrostatic constant (8.99 × 10^9 N m^2/C^2).
2. The x-component of the electric field due to q1 at point P can be calculated as Ex1 = (k * q1) / (r1^2), where r1 is the distance from q1 to P. Since the triangle is equilateral, the distance r1 is equal to half the length of AB, which is 0.46 m.
3. Substitute the values and calculate Ex1.
b) The y-component of the electric field at point P:
1. Repeat the same procedure as above, but this time calculate the y-component of the electric field due to each charge at point P.
2. The distance from q1 to P, r1, remains the same as before.
3. Calculate Ey1, Ey2, and Ey3 for charges q1, q2, and q3 respectively.
4. The total y-component of the electric field is given by Ey = Ey1 + Ey2 + Ey3.
c) The electric potential at point P:
1. The electric potential at a point due to a point charge is given by the formula V = k * q / r, where r is the distance from the point charge to the point where the potential is calculated.
2. Calculate the electric potential at point P due to each charge individually using the formula V = k * q / r.
3. The total electric potential at point P, Vp, is calculated by summing the potentials due to each charge.
d) The electric potential energy of the system:
1. The electric potential energy of a system of charges is given by the formula U = k * q1 * q2 / r12 + k * q1 * q3 / r13 + k * q2 * q3 / r23, where r12, r13, and r23 are the distances between pairs of charges.
2. Calculate the potential energy of the system using the above formula and substituting the given values for charges q1, q2, and q3, and the distances between them.
By following these steps, you can find the x-component and y-component of the electric field at point P, the electric potential at point P, and the electric potential energy of the system.