A line of charge with a uniform density of 36.3 nC/m lies along the line y = -14.8 cm, between the points with coordinates x = 0 and x = 44.9 cm. Calculate the electric field it creates at the origin. Enter first the x- and then the y-component.
To calculate the electric field at the origin due to the line of charge, we can use the principle of superposition. We'll divide the line of charge into infinitesimally small segments, calculate the contribution of each segment to the electric field at the origin, and then integrate over the entire length of the line.
The electric field at a point due to a small segment of charge is given by Coulomb's Law:
dE = kdq / r^2
Where:
- dE is the electric field due to the small segment,
- k is the Coulomb's constant (8.99 x 10^9 N m^2/C^2),
- dq is the charge of the small segment,
- r is the distance between the small segment and the point at which we want to find the electric field.
To find the x- and y-components of the electric field, we need to break down the electric field contributions from each small segment into their x- and y-components. The x-component of the elemental electric field dE_x can be expressed as dE * cos(θ), and the y-component dE_y can be expressed as dE * sin(θ), where θ is the angle between the direction of the electric field and the positive x-axis.
We can express dq, the charge of each small segment of the line of charge, in terms of the linear charge density λ (given in nC/m) and the length of each segment dl. The relationship is dq = λ * dl.
To find the electric field at the origin, we can integrate the contributions from each small segment over the entire length of the line of charge. Since the line of charge lies along the y-axis (y = -14.8 cm), the x-component of the electric field due to each segment will be cancelled out by the x-component of the electric field from a segment at the same distance but on the other side of the y-axis. Therefore, we only need to consider the y-component of the electric field, which will be additive.
Let's proceed with the calculation:
1. Calculate the y-component of the electric field due to a small segment at a distance x from the origin:
dE_y = (k * λ * dl * sin(θ)) / r^2
2. Express dl in terms of dx (Note: since the line of charge lies along the y-axis, dl = dy):
dl = dy
3. Express θ in terms of x and y:
θ = atan(y / x)
4. Express r in terms of x:
r = √(x^2 + y^2)
5. Express λ in terms of the charge density σ:
λ = σ * dy = (σ/100) * dx (Note: σ is given in nC/cm, so we need to convert it to nC/m by dividing by 100)
6. Substitute the above expressions into the equation for dE_y:
dE_y = (k * (σ/100) * dx * sin(atan(y / x))) / (x^2 + y^2)
7. Integrate the expression for dE_y over the entire length of the line of charge (from x = 0 to x = 44.9 cm) to calculate the electric field at the origin:
Ey = ∫[from 0 to 0.449 m] (k * (σ/100) * dx * sin(atan(- 0.148 / x))) / (x^2 + (-0.148)^2)
Evaluate this integral to find the y-component of the electric field at the origin. Then, the x-component will be zero since the line of charge lies along the y-axis.
To find the electric field created by the line of charge at the origin, we can use Coulomb's law. The electric field at a point due to a continuous charge distribution is given by the integral of the electric field contribution from each element of charge.
First, let's calculate the electric field contribution from a small element of charge with length dx at a distance x from the origin.
The charge density, λ, is given by 36.3 nC/m.
The electric field contribution, dE, due to a small element of charge is given by:
dE = (k * dq) / r^2
where k is the electrostatic constant (9 × 10^9 N m^2/C^2), dq is the charge of the small element of charge, and r is the distance from the small element to the origin.
The small element of charge dq is given by dq = λ * dx.
We can calculate the electric field, dEx, at the origin due to the x-component of the electric field contribution:
dEx = (k * dq * cosθ) / r^2
where θ is the angle between the x-component of the electric field and the x-axis.
Since the line of charge is parallel to the y-axis, the angle θ is 90 degrees, and cosθ = 0.
Therefore, the x-component of the electric field, dEx, is 0.
The y-component of the electric field, dEy, at the origin due to the y-component of the electric field contribution is given by:
dEy = (k * dq * sinθ) / r^2
where sinθ = 1 (since the angle θ is 90 degrees).
Now, let's calculate the total electric field at the origin by integrating the contributions from all the small elements of charge.
Etotal = ∫ dE
To integrate, we need to express dEy in terms of x.
The distance r from a small element of charge at position x to the origin is given by:
r = sqrt(x^2 + y^2)
Since we are calculating the electric field at the origin, y = -14.8 cm = -0.148 m.
Therefore, r = sqrt(x^2 + (-0.148)^2)
We can substitute dq = λ * dx and sinθ = 1 into the equation for dEy:
dEy = (k * λ * dx) / (x^2 + 0.148^2)
Now, we can find the total electric field at the origin by integrating dEy over the range of x from 0 to 44.9 cm = 0.449 m.
Etotal = ∫ dEy = ∫ [k * λ * dx / (x^2 + 0.148^2)]
Etotal = k * λ * [ln(x^2 + 0.148^2)] evaluated from 0 to 0.449
Etotal = k * λ * [ln(0.449^2 + 0.148^2) - ln(0.148^2)]
Finally, we can substitute the values of k, λ, and calculate the electric field at the origin:
k = 9 × 10^9 N m^2/C^2
λ = 36.3 × 10^(-9) C/m
Etotal = (9 × 10^9 N m^2/C^2) * (36.3 × 10^(-9) C/m) * [ln(0.449^2 + 0.148^2) - ln(0.148^2)]
Calculating the above expression will give us the electric field at the origin, with the x-component being 0 and the y-component being the final value.