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Particle in electric field of an infinite cylinder
Description
In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:
| $\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$ |
For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:
| $ S =2 \pi r h $ |
what is shown in the graph
Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
ID:(11838, 0)
Particle in electric potencial of an infinite cylinder
Description
In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:
| $\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$ |
For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:
| $ S =2 \pi r h $ |
Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:
| $ \varphi_c = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)$ |
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:
| $ \varphi_1 = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r_1 }{ r_0 }\right)$ |
and the electric potential 2 ($\varphi_2$), according to the equation:
| $ \varphi_2 = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r_2 }{ r_0 }\right)$ |
must satisfy the following relationship:
| $ \displaystyle\frac{1}{2} m v_1 ^2 + q \varphi_1 = \displaystyle\frac{1}{2} m v_2 ^2 + q \varphi_2 $ |
ID:(11845, 0)
Conducting cylinder
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Equations
In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:
| $\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$ |
For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:
| $ S =2 \pi r h $ |
Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
(ID 11445)
In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:
| $\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$ |
For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:
| $ S =2 \pi r h $ |
Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
(ID 11445)
The reference electrical, infinite conducting cylinder ($\varphi_c$) is derived from the radial integration of the electric field, infinite conducting cylinder ($E_c$) from the cylinder radius ($r_0$) to the axle distance ($r$), resulting in the following equation:
| $ \varphi_c = -\displaystyle\int_{r_0}^r du E_c$ |
Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field, infinite conducting cylinder ($E_c$) is given as:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
This implies that by performing the integration
$\varphi_c = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$
the following equation is obtained:
| $ \varphi_c = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)$ |
(ID 11585)
The reference electrical, infinite conducting cylinder ($\varphi_c$) is derived from the radial integration of the electric field, infinite conducting cylinder ($E_c$) from the cylinder radius ($r_0$) to the axle distance ($r$), resulting in the following equation:
| $ \varphi_c = -\displaystyle\int_{r_0}^r du E_c$ |
Furthermore, for the variables the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field, infinite conducting cylinder ($E_c$) is given as:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
This implies that by performing the integration
$\varphi_c = -\displaystyle\int_{r_0}^r du \displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon u }= -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon } \ln\left(\displaystyle\frac{ r }{ r_0 }\right)$
the following equation is obtained:
| $ \varphi_c = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)$ |
(ID 11585)
Examples
(ID 15794)
In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:
| $\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$ |
For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:
| $ S =2 \pi r h $ |
what is shown in the graph
Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
(ID 11838)
In the case of a cylindrical Gaussian surface, the electric field ($\vec{E}$) is constant in the direction of the versor normal to the section ($\hat{n}$). Therefore, using the variables the charge ($Q$), the electric field constant ($\epsilon_0$), and the dielectric constant ($\epsilon$), the integral over the surface where the electric field is constant ($dS$) can be calculated through the following equation:
| $\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$ |
For a cylinder characterized by the axle distance ($r$) and the conductor length ($L$), the following applies:
| $ S =2 \pi r h $ |
Furthermore, the linear charge density ($\lambda$) is calculated using the charge ($Q$) according to the equation:
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
Thus, it is established that the electric field, infinite conducting cylinder ($E_c$) is:
| $ \varphi_c = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)$ |
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:
| $ \varphi_1 = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r_1 }{ r_0 }\right)$ |
and the electric potential 2 ($\varphi_2$), according to the equation:
| $ \varphi_2 = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r_2 }{ r_0 }\right)$ |
must satisfy the following relationship:
| $ \displaystyle\frac{1}{2} m v_1 ^2 + q \varphi_1 = \displaystyle\frac{1}{2} m v_2 ^2 + q \varphi_2 $ |
(ID 11845)
(ID 15804)
The linear charge density ($\lambda$) is calculated as the charge ($Q$) divided by the conductor length ($L$):
| $ \lambda = \displaystyle\frac{ Q }{ L }$ |
(ID 11459)
The electric field, infinite conducting cylinder ($E_c$) is with the pi ($\pi$), the electric field constant ($\epsilon_0$), the dielectric constant ($\epsilon$), the linear charge density ($\lambda$) and the axle distance ($r$) is equal to:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
(ID 11445)
The electric field, infinite conducting cylinder ($E_c$) is with the pi ($\pi$), the electric field constant ($\epsilon_0$), the dielectric constant ($\epsilon$), the linear charge density ($\lambda$) and the axle distance ($r$) is equal to:
| $ E_c =\displaystyle\frac{1}{2 \pi \epsilon_0 \epsilon }\displaystyle\frac{ \lambda }{ r }$ |
(ID 11445)
The reference electrical, infinite conducting cylinder ($\varphi_c$) 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 cylinder radius ($r_0$) is equal to:
| $ \varphi_c = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)$ |
(ID 11585)
The reference electrical, infinite conducting cylinder ($\varphi_c$) 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 cylinder radius ($r_0$) is equal to:
| $ \varphi_c = -\displaystyle\frac{ \lambda }{ 2 \pi \epsilon_0 \epsilon }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)$ |
(ID 11585)
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:
| $ \displaystyle\frac{1}{2} m v_1 ^2 + q \varphi_1 = \displaystyle\frac{1}{2} m v_2 ^2 + q \varphi_2 $ |
(ID 11596)
ID:(2075, 0)