find the electrostatic potential,
and then the electric field a distance z above the centre of a circular loop of radius r, which carries a uniform line charge (lamda)
To find the electrostatic potential and electric field at a distance z above the center of a circular loop carrying a uniform line charge (lambda), you can follow these steps:
1. Start by considering a small element of the circular loop, which is of length dl and carries a small charge deltaq. Let's denote the angle between the line connecting the small element to the point of interest and the vertical axis as theta.
2. The electric potential (V) at the point of interest due to this small segment is given by the formula:
dV = k * deltaq / r
where k is the electrostatic constant (9 x 10^9 Nm^2/C^2) and r is the distance from the small element to the point of interest.
3. Express deltaq in terms of lambda and dl:
deltaq = lambda * dl
4. Express r in terms of z and the radius of the circular loop (r):
r = sqrt(r^2 + z^2)
5. Substitute lambda * dl for deltaq and sqrt(r^2 + z^2) for r in the formula for dV:
dV = k * lambda * dl / sqrt(r^2 + z^2)
6. Integrate over the entire circular loop to obtain the total electric potential (V):
V = ∫ dV
7. To find the electric field (E), take the negative gradient of the electrostatic potential:
E = -∇V,
where ∇ is the del operator.
Alternatively, if you already know the electrostatic potential V(r) due to a circular loop carrying line charge, you can directly find the electric field by taking the negative gradient of the potential:
E = -∇V
where the del operator in cylindrical coordinates is given by:
∇ = (∂/∂r) * er + (1/r) * (∂/∂θ) * eθ + (∂/∂z) * ez
and er, eθ, and ez are unit vectors in the radial, azimuthal, and vertical directions, respectively.