proof formula for charge line disk and ring in both electric field and potential
To derive the formulas for the electric field and electric potential of a charged line, disk, or ring, we'll discuss each case separately.
1. Charged Line:
A charged line refers to an infinitely long wire with charge uniformly distributed along its length.
a) Electric Field:
To find the electric field at a point P located at a distance r from the line, we can use Gauss's Law. By considering a cylindrical Gaussian surface centered on the line and applying symmetry arguments, we can prove that the electric field is directly proportional to the linear charge density (λ) of the line. The formula for the electric field (E) for a charged line is:
E = k * λ / r
Where:
- E is the electric field strength at a distance r from the line.
- k is Coulomb's constant (9 x 10^9 Nm^2/C^2).
- λ is the linear charge density (C/m) of the line.
b) Electric Potential:
The electric potential (V) at a point P located at a distance r from the line can be obtained by integrating the electric field formula. The formula for the electric potential of a charged line is:
V = k * λ * ln(λr0) / 2π
Where:
- V is the electric potential at a distance r from the line.
- k is Coulomb's constant.
- λ is the linear charge density of the line.
- r0 is a reference distance often taken as 1 meter.
2. Charged Disk:
A charged disk refers to a circular disk with charge uniformly distributed over its surface. To calculate the electric field and potential, we assume that the point of interest lies on the axis of the disk.
a) Electric Field:
The electric field at a point P, located at a distance z from the center of the disk, can be found by considering elemental rings on the disk's surface and integrating their contributions. The formula for the electric field is:
E = (k * σ * z) / (2ε₀) * [1 / sqrt(R^2 + z^2) - 1 / sqrt(R^2 + h^2)]
Where:
- E is the electric field strength at a distance z from the center of the disk.
- k is Coulomb's constant.
- σ is the surface charge density (C/m²) of the disk.
- R is the radius of the disk.
- ε₀ is the electric constant.
b) Electric Potential:
To calculate the electric potential at a point P located at a distance z from the center of the disk, we integrate the previously derived electric field formula. The electric potential (V) for a charged disk is:
V = (k * σ / ε₀) * [sqrt(R^2 + z^2) - sqrt(R^2 + h^2)]
Where:
- V is the electric potential at a distance z from the center of the disk.
- k is Coulomb's constant.
- σ is the surface charge density of the disk.
- R is the radius of the disk.
- ε₀ is the electric constant.
3. Charged Ring:
A charged ring refers to a thin circular ring with charge uniformly distributed along its circumference.
a) Electric Field:
The electric field at a point P located on the axis of the ring can be calculated using Coulomb's Law and symmetry arguments. The formula for the electric field is:
E = (k * Q * z) / (2πε₀ * (R^2 + z^2)^(3/2))
Where:
- E is the electric field strength at a distance z from the center of the ring.
- k is Coulomb's constant.
- Q is the total charge of the ring.
- R is the radius of the ring.
- ε₀ is the electric constant.
b) Electric Potential:
The electric potential at a point P located on the axis of the ring can be obtained by integrating the previously derived electric field formula. The electric potential (V) for a charged ring is:
V = (k * Q) / (sqrt(R^2 + z^2))
Where:
- V is the electric potential at a distance z from the center of the ring.
- k is Coulomb's constant.
- Q is the total charge of the ring.
- R is the radius of the ring.
Note: It's important to use consistent units when plugging values into these formulas and be aware of the assumptions made in these derivations, such as the assumptions of symmetry or infinite length.