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Wire
Storyboard 
The geometry referred to as a wire can be understood as an infinitely long cylinder where the distance to the axis is much greater than the radius of the cylinder. Essentially, this corresponds to a case where the radius approaches zero, effectively becoming an infinitely thin line of charge.
ID:(2073, 0)
Wire
Storyboard
The geometry referred to as a wire can be understood as an infinitely long cylinder where the distance to the axis is much greater than the radius of the cylinder. Essentially, this corresponds to a case where the radius approaches zero, effectively becoming an infinitely thin line of charge.
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In the case of a spherical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), it can be calculated by integrating over the surface where the electric field is constant ($dS$):
With the surface ($S$) for a cylinder of the axle distance ($r$) and the conductor length ($L$):
and the linear charge density ($\lambda$) calculated with the charge ($Q$):
Thus,
In the case of a spherical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), it can be calculated by integrating over the surface where the electric field is constant ($dS$):
With the surface ($S$) for a cylinder of the axle distance ($r$) and the conductor length ($L$):
and the linear charge density ($\lambda$) calculated with the charge ($Q$):
Thus,
The electric potential, infinite wire ($\varphi_w$) is derived from the radial integration of the electric field of an infinite wire ($E_w$) from the reference radius ($r_0$) to the axle distance ($r$), resulting in the following equation:
Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field of an infinite wire ($E_w$) is given as:
This implies that by performing the integration
$\varphi_w = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$
the following equation is obtained:
The electric potential, infinite wire ($\varphi_w$) is derived from the radial integration of the electric field of an infinite wire ($E_w$) from the reference radius ($r_0$) to the axle distance ($r$), resulting in the following equation:
Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field of an infinite wire ($E_w$) is given as:
This implies that by performing the integration
$\varphi_w = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$
the following equation is obtained:
Examples
In the case of a spherical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), it can be calculated by integrating over the surface where the electric field is constant ($dS$):
With the surface ($S$) for a cylinder of the axle distance ($r$) and the conductor length ($L$):
what is shown in the graph
and the linear charge density ($\lambda$) calculated with the charge ($Q$):
Thus,
The electric potential, infinite wire ($\varphi_w$) is derived from the radial integration of the electric field of an infinite wire ($E_w$) from the reference radius ($r_0$) to the axle distance ($r$), resulting in the following equation:
Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field of an infinite wire ($E_w$) is given as:
This implies that by performing the integration
$\varphi_w = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$
the following equation is obtained:
As illustrated in the following graph:
the field at two points must have the same energy. Therefore, the variables the charge ($Q$), the particle mass ($m$), the speed 1 ($v_1$), the speed 2 ($v_2$), and the electric potential 1 ($\varphi_1$) according to the equation:
and the electric potential 2 ($\varphi_2$), according to the equation:
must satisfy the following relationship:
The electric field of an infinite wire ($E_w$) is a function of the linear charge density ($\lambda$), the axle distance ($r$), the dielectric constant ($\epsilon$) and the electric field constant ($\epsilon_0$) and is calculated through:
The linear charge density ($\lambda$) is calculated as the charge ($Q$) divided by the conductor length ($L$):
The electric field of an infinite wire ($E_w$) is a function of the linear charge density ($\lambda$), the axle distance ($r$), the dielectric constant ($\epsilon$) and the electric field constant ($\epsilon_0$) and is calculated through:
The electric potential, infinite wire ($\varphi_w$) is with the pi ($\pi$), the electric field constant ($\epsilon_0$), the dielectric constant ($\epsilon$), the linear charge density ($\lambda$), the axle distance ($r$) and the reference radius ($r_0$) is equal to:
The electric potential, infinite wire ($\varphi_w$) is with the pi ($\pi$), the electric field constant ($\epsilon_0$), the dielectric constant ($\epsilon$), the linear charge density ($\lambda$), the axle distance ($r$) and the reference radius ($r_0$) is equal to:
Electric potentials, which represent potential energy per unit of charge, influence how the velocity of a particle varies. Consequently, due to the conservation of energy between two points, it follows that in the presence of variables the charge ($q$), the particle mass ($m$), the speed 1 ($v_1$), the speed 2 ($v_2$), the electric potential 1 ($\varphi_1$), and the electric potential 2 ($\varphi_2$), the following relationship must be satisfied:
ID:(2073, 0)
