For any given quadrupole, let the charges be +q, −q, +q, and −q at the corners of the square, and l be the side of the square. Calculate approximately the electric potential and the electric field far away from the quadrupole, IN THE THREE-DIMENSIONAL SPACE. You will verify that φ decays like 1/r^3 and E decays like 1/r^4 at large r, where r is the distance between the point of interest and the center of the square. Assume that r >> l.
To calculate the electric potential and electric field of a quadrupole in three-dimensional space, we can use the superposition principle. This principle states that the total potential or field at a point due to a collection of charges is the sum of the potentials or fields produced by each individual charge.
Let's start by calculating the electric potential at a point far away from the quadrupole. To do this, we can approximate the potential as the sum of the potentials produced by each charge:
φ = φ1 + φ2 + φ3 + φ4
where φ1, φ2, φ3, and φ4 are the potentials produced by the four charges +q, -q, +q, and -q, respectively.
The potential at a point due to a point charge q can be calculated using the equation:
φ = k * q / r
where k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2) and r is the distance between the charge and the point of interest.
For the quadrupole, let's assume that the center of the square is the origin of our coordinate system. Since we are interested in points far away from the quadrupole (r >> l), we can consider only the leading term in the potential expression.
For the charge +q at one of the corners of the square, the distance from this charge to the point of interest is approximately r. Thus, the potential produced by this charge is given by:
φ1 = k * q / r
Similarly, the potentials produced by the other charges can also be approximated as φ2 = -k * q / r, φ3 = k * q / r, and φ4 = -k * q / r.
Adding all these potentials together, we get:
φ = (k * q / r) - (k * q / r) + (k * q / r) - (k * q / r) = 0
Therefore, the electric potential at a point far away from the quadrupole is approximately zero.
Now let's move on to calculating the electric field. The electric field is the negative gradient of the potential:
E = -∇φ
where ∇φ is the gradient of the potential. In three-dimensional space, the gradient is given by:
∇φ = (∂φ/∂x, ∂φ/∂y, ∂φ/∂z)
Since our potential is 0, the gradient is also zero, which means that the electric field far away from the quadrupole is approximately zero as well.
Therefore, the electric potential and electric field both decay as 1/r^3 at large distances (r >> l) from the quadrupole.
To calculate the electric potential and electric field far away from the quadrupole, we can use the principle of superposition. We know that the electric potential at a point due to a charge q is given by:
V = k*q/r
Where V is the potential, k is the electrostatic constant (k = 9 x 10^9 Nm^2/C^2), q is the charge, and r is the distance from the charge.
Since the electric potential is a scalar quantity, we can add up the potentials from each of the four charges in the quadrupole to find the total potential. Taking into account the signs of the charges and the distances, the total potential at a point far away from the quadrupole is:
V_total = k*q/r + (-k*q/r) + k*q/r + (-k*q/r)
= 0
Therefore, the electric potential far away from the quadrupole is approximately zero.
Now, let's calculate the electric field far away from the quadrupole. The electric field is the negative gradient of the electric potential, so we can differentiate the potential expression to find the electric field:
E = -dV/dr
Taking the derivative, we get:
E = -k*q/r^2 + k*q/r^2 - k*q/r^2 + k*q/r^2
= 0
Therefore, the electric field far away from the quadrupole is approximately zero.
In conclusion, both the electric potential and the electric field are approximately zero far away from the quadrupole in the three-dimensional space. This result verifies that the potential (φ) decays like 1/r^3 and the electric field (E) decays like 1/r^4 at large distances (r >> l), where r is the distance between the point of interest and the center of the square.