The three charges shown in Figure P16.17 are at the vertices of an isosceles triangle. If q = 16.0 10-9 C, calculate the electric potential at the midpoint of the base.
V
Figure P16.17
r1 =r3 = 1cm = 10^-2 m
The Pythagorean Theorem gives the distance from the midpoint of the base to the charge at the apex of the triangle as
r2 = sqrt(4² - 1²) = sqrt(15) = 3.87 cm = 3.87•10^-2 m.
The potential at the midpoint of the base is
φ = k•q1/r1 + k•q2/r2 + k•q3/r3 = k{(q1+q3)/r1+ q2/r2} =
= 9•10^9• {(- 16•10^-9 - 16•10^-9)/10^-2 +16•10^-9/3.87•10^-2} =
= 9•10^9•(-32•10^-7 +4.13•10^-7)= - 2.5•10^5 V = - 25 kV
To calculate the electric potential at the midpoint of the base of the isosceles triangle, we need to find the sum of the electric potentials due to each individual charge at that point.
First, let's label the charges in the figure as q1, q2, and q3, with q = 16.0 * 10^(-9) C.
The electric potential due to a point charge at a distance r from the charge can be calculated using the formula:
V = k * q / r
where V is the electric potential, k is Coulomb's constant (k = 9 * 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge.
Now, let's calculate the electric potentials due to each charge at the midpoint of the base.
Charge q1 is at the top vertex of the triangle. Since the distance from q1 to the midpoint of the base is equal to the length of the base divided by 2, we can calculate the electric potential due to q1 as:
V1 = k * q1 / (length/2)
Charge q2 is at the bottom vertex of the triangle, which is also the midpoint of the base. Since the distance from q2 to the midpoint of the base is 0, the electric potential due to q2 is:
V2 = k * q2 / 0
It is important to note that dividing by 0 is undefined in mathematics. In this case, it means that the potential due to q2 is infinite.
Charge q3 is at one of the other vertices of the triangle. The distance from q3 to the midpoint of the base can be calculated using the Pythagorean theorem, since it forms a right triangle with the base. Let's call that distance d. Then, the electric potential due to q3 is:
V3 = k * q3 / d
Finally, to find the total electric potential at the midpoint of the base, we can add up V1, V2, and V3:
V_total = V1 + V2 + V3
However, since V2 is undefined (infinite), the total potential will also be infinite.
Therefore, the electric potential at the midpoint of the base of the isosceles triangle is infinite.