An electric dipole is formed from two charges, ±q, spaced 1.00cm apart. The dipole is at the origin, oriented along the y-axis. The electric field strength at the point (x,y)=(0cm,10cm) is 320N/C .
What is the charge q? Give your answer in nC.
What is the electric field strength at the point (x,y)=(10cm,0cm)?
To find the charge, we need to use the formula for the electric field strength of a dipole:
E = k * |p| / r^3,
where E is the electric field strength, k is the electrostatic constant (k ≈ 9 × 10^9 N m^2 / C^2), |p| is the magnitude of the electric dipole moment, and r is the distance between the point and the center of the dipole.
First, let's find the magnitude of the electric dipole moment |p|:
|p| = q * d,
where q is the charge and d is the distance between the charges.
We know that the distance between the charges is 1.00 cm, so |p| = q * 0.01 m.
Now let's find the distance between the point (0 cm, 10 cm) and the center of the dipole:
r = sqrt(x^2 + y^2) = sqrt(0^2 + 0.1^2) = 0.1 m.
Now we can plug in the known values into the formula for the electric field strength:
320 N/C = (9 × 10^9 N m^2 / C^2) * (q * 0.01 m) / (0.1 m)^3.
Solving for q, we get:
q ≈ 1.07 nC.
Now let's find the electric field strength at the point (x, y) = (10 cm, 0 cm).
First, let's find the distance between the point and the center of the dipole:
r = sqrt(x^2 + y^2) = sqrt(0.1^2 + 0^2) = 0.1 m.
Now we can plug in the known values and the charge q into the formula for the electric field strength:
E = (9 × 10^9 N m^2 / C^2) * (1.07 nC * 0.01 m) / (0.1 m)^3.
Calculating the electric field strength, we get:
E ≈ 32 N/C.
To find the charge q, we can use the formula for the electric field due to an electric dipole at a point on its axial line:
E = (k * 2*q * d) / (r^2 + (d/2)^2) ------------(1)
where E is the electric field strength, k is the electrostatic constant (8.99 × 10^9 N m^2/C^2), q is the charge, d is the distance between the charges, and r is the distance from the midpoint of the dipole to the point where we want to find the electric field.
Given values:
E = 320 N/C
d = 1.00 cm = 0.01 m
r = 10 cm = 0.1 m
Substituting these values into equation (1):
320 = (8.99 × 10^9 * 2q * 0.01) / ((0.1)^2 + (0.01/2)^2)
Simplifying the equation:
320 = (17.98 × 10^7 * q) / (0.01 + 0.0001/4)
320 = (17.98 × 10^7 * q) / 0.0101
Solving for q:
q = (320 * 0.0101) / (17.98 × 10^7)
= 0.0032 C
Converting C to nC:
q = 0.0032 C * 10^9
= 3200 nC
Therefore, the value of the charge q is 3200 nC.
To find the electric field strength at the point (x,y) = (10 cm, 0 cm), we can use the formula for the electric field due to an electric dipole at a point on its equatorial line:
E' = (k * q * d * x) / (r^3) --------------(2)
where E' is the electric field strength, k is the electrostatic constant, q is the charge, d is the distance between the charges, x is the distance from the dipole to the point along the x-axis, and r is the distance from the midpoint of the dipole to the point where we want to find the electric field.
Given values:
q = 3200 nC = 3200 × 10^-9 C
d = 1.00 cm = 0.01 m
x = 10 cm = 0.1 m
r = 10 cm = 0.1 m
Substituting these values into equation (2):
E' = (8.99 × 10^9 * 3200 × 10^-9 * 0.01 * 0.1) / (0.1)^3
Simplifying the equation:
E' = (8.99 × 10^9 * 3200 × 10^-9 * 0.01 * 0.1) / 0.001
E' = 8.99 × 3200 × 10^-9
E' = 28.768 N/C
Therefore, the electric field strength at the point (x,y) = (10 cm, 0 cm) is 28.768 N/C.
To find the charge q, we can use the concept of electric dipole moment. The electric dipole moment is given by the formula:
p = qd
Where p is the dipole moment, q is the magnitude of the charge, and d is the distance between the charges.
In this case, the dipole moment p is given as the product of the electric field strength E and the distance between the charges d:
p = Ed
Since the electric field strength is given as 320 N/C and the distance between the charges is given as 1.00 cm (which is equal to 0.01 m), we can substitute these values into the equation:
p = (320 N/C)(0.01 m) = 3.2 Nm
Since the dipole moment is measured in Coulomb meters (Cm), we can determine the charge q by rearranging the equation:
q = p/d
q = (3.2 Nm) / (0.01 m) = 320 nC
Therefore, the charge q is 320 nC.
To find the electric field strength at the point (x,y) = (10 cm, 0 cm), we can use the formula for the electric field due to a dipole at any point in space:
E = (k * (2p * y) / r^3) * i
Where E is the electric field strength, k is the electrostatic constant (9 × 10^9 Nm^2/C^2), p is the dipole moment, y is the distance (or displacement) along the y-axis from the dipole, r is the distance from the dipole to the point, and i is the unit vector in the x-direction.
In this case, the dipole moment p is already known to be 3.2 Nm and the distance (or displacement) along the y-axis y is given as 0 cm (which is equal to 0 m). The distance r can be calculated using the Pythagorean theorem:
r = sqrt((x - 0)^2 + (y - d)^2)
Here, x = 10 cm (which is equal to 0.1 m) and d = 1 cm (which is equal to 0.01 m). Substituting these values into the equation, we get:
r = sqrt((0.1 m)^2 + (0 - 0.01 m)^2) = 0.1005 m
Now we can substitute all the known values into the formula for the electric field:
E = (9 × 10^9 Nm^2/C^2) * (2 * 3.2 Nm * 0 m) / ((0.1005 m)^3) * i
Calculating the numerical value:
E ≈ 57.14 N/C
Therefore, the electric field strength at the point (x,y) = (10 cm, 0 cm) is approximately 57.14 N/C.