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A point charge is an idealized model in physics where a charge is concentrated at a single point with no dimensions. It generates an electric field that radiates uniformly outward, decreasing in strength with the square of the distance.

>Model

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Iframe

>Top

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Concept

>Top

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 sphere of radius a distance between charges ($r$):

Thus, the electric field of a point charge ($E_p$) results in:

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Concept

>Top

The electric potential, point charge ($\varphi_p$) is calculated from the radial integration of the electric field of a point charge ($E_p$) from the radius ($r$) to infinity, which results in

On the other hand, for the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field of a point charge ($E_p$) is

This implies that by integrating

$\varphi_p = -\displaystyle\int_{r}^{\infty} du \displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon u^2 }= -\displaystyle\frac{ Q }{ 4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

we obtain

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:

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Top

>Top

Parameters

$\epsilon$

epsilon

Dielectric constant

-

$\epsilon_0$

epsilon_0

Electric field constant

C^2/m^2N

$m$

m

Particle mass

kg

$\pi$

pi

Pi

rad

Variables

$Q$

Q

Charge

C

$E_{p1}$

E_p1

Electric field of a point charge in 1

V/m

$E_{p2}$

E_p2

Electric field of a point charge in 2

V/m

$\varphi_1$

phi_1

Electric potential 1

V

$\varphi_2$

phi_2

Electric potential 2

V

$r_1$

r_1

Radius 1

m

$r_2$

r_2

Radius 2

m

$v_1$

v_1

Speed 1

m/s

$v_2$

v_2

Speed 2

m/s

$q$

q

Test charge

C

Calculations

• Alternative method
• Traditional method
• Strategy
• 2D
• 3D

Equation

Number

First, select the equation: to , then, select the variable: to

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Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable Given Calculate Target : Equation To be used

Equations

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Equation

>Top, >Model

The electric field of a point charge ($E_p$) is a function of the charge ($Q$), the electric field constant ($\epsilon_0$), the dielectric constant ($\epsilon$), and the distance between charges ($r$) and is calculated as follows:

$E_{p1} =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r_1 ^2}$

$E_p =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r ^2}$

Conditions

Variables

$Q$

Charge

$C$

5459

$\epsilon$

Dielectric constant

$-$

5463

$r$

$r_1$

Radius 1

$m$

10390

$\epsilon_0$

Electric field constant

8.854187e-12

$C^2/m^2N$

5462

$E_p$

$E_{p1}$

Electric field of a point charge in 1

$V/m$

10474

$\pi$

Pi

3.1415927

$rad$

5057

Relation

Development

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 ($dS$):

$\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$

with the surface ($S$) for a sphere of radius a distance between charges ($r$):

$S = 4 \pi r ^2$

Thus, the electric field of a point charge ($E_p$) results in:

$E_p =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r ^2}$

ID:(11442, 1)

Equation

>Top, >Model

The electric field of a point charge ($E_p$) is a function of the charge ($Q$), the electric field constant ($\epsilon_0$), the dielectric constant ($\epsilon$), and the distance between charges ($r$) and is calculated as follows:

$E_{p2} =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r_2 ^2}$

$E_p =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r ^2}$

Conditions

Variables

$Q$

Charge

$C$

5459

$\epsilon$

Dielectric constant

$-$

5463

$r$

$r_2$

Radius 2

$m$

10391

$\epsilon_0$

Electric field constant

8.854187e-12

$C^2/m^2N$

5462

$E_p$

$E_{p2}$

Electric field of a point charge in 2

$V/m$

10475

$\pi$

Pi

3.1415927

$rad$

5057

Relation

Development

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 ($dS$):

$\displaystyle\int_S\vec{E}\cdot\hat{n}\,dS=\displaystyle\frac{Q}{\epsilon_0\epsilon}$

with the surface ($S$) for a sphere of radius a distance between charges ($r$):

$S = 4 \pi r ^2$

Thus, the electric field of a point charge ($E_p$) results in:

$E_p =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r ^2}$

ID:(11442, 2)

Equation

>Top, >Model

The electric potential, point charge ($\varphi_p$) is with the charge ($Q$), the distance between charges ($r$), the dielectric constant ($\epsilon$) and the electric field constant ($\epsilon_0$) equal to:

$\varphi_1 = -\displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r_1 }$

$\varphi_p = -\displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

Conditions

Variables

$Q$

Charge

$C$

5459

$\epsilon$

Dielectric constant

$-$

5463

$r$

$r_1$

Radius 1

$m$

10390

$\epsilon_0$

Electric field constant

8.854187e-12

$C^2/m^2N$

5462

$\varphi_p$

$\varphi_1$

Electric potential 1

$V$

10392

$\pi$

Pi

3.1415927

$rad$

5057

Relation

Development

The electric potential, point charge ($\varphi_p$) is calculated from the radial integration of the electric field of a point charge ($E_p$) from the radius ($r$) to infinity, which results in

$\varphi_p = -\displaystyle\int_r^{\infty} du\,E_p$

On the other hand, for the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field of a point charge ($E_p$) is

$E_p =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r ^2}$

This implies that by integrating

$\varphi_p = -\displaystyle\int_{r}^{\infty} du \displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon u^2 }= -\displaystyle\frac{ Q }{ 4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

we obtain

$\varphi_p = -\displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

ID:(11576, 1)

Equation

>Top, >Model

The electric potential, point charge ($\varphi_p$) is with the charge ($Q$), the distance between charges ($r$), the dielectric constant ($\epsilon$) and the electric field constant ($\epsilon_0$) equal to:

$\varphi_2 = -\displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r_2 }$

$\varphi_p = -\displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

Conditions

Variables

$Q$

Charge

$C$

5459

$\epsilon$

Dielectric constant

$-$

5463

$r$

$r_2$

Radius 2

$m$

10391

$\epsilon_0$

Electric field constant

8.854187e-12

$C^2/m^2N$

5462

$\varphi_p$

$\varphi_2$

Electric potential 2

$V$

10393

$\pi$

Pi

3.1415927

$rad$

5057

Relation

Development

The electric potential, point charge ($\varphi_p$) is calculated from the radial integration of the electric field of a point charge ($E_p$) from the radius ($r$) to infinity, which results in

$\varphi_p = -\displaystyle\int_r^{\infty} du\,E_p$

On the other hand, for the charge ($Q$), the dielectric constant ($\epsilon$), and the electric field constant ($\epsilon_0$), the value of the electric field of a point charge ($E_p$) is

$E_p =\displaystyle\frac{1}{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{Q}{ r ^2}$

This implies that by integrating

$\varphi_p = -\displaystyle\int_{r}^{\infty} du \displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon u^2 }= -\displaystyle\frac{ Q }{ 4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

we obtain

$\varphi_p = -\displaystyle\frac{ Q }{4 \pi \epsilon_0 \epsilon }\displaystyle\frac{1}{ r }$

ID:(11576, 2)

Equation

>Top, >Model

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:

$\displaystyle\frac{1}{2} m v_1 ^2 + q \varphi_1 = \displaystyle\frac{1}{2} m v_2 ^2 + q \varphi_2$

Conditions

Variables

$\varphi_1$

Electric potential 1

$V$

10392

$\varphi_2$

Electric potential 2

$V$

10393

$m$

Particle mass

$kg$

5516

$v_1$

Speed 1

$m/s$

8562

$v_2$

Speed 2

$m/s$

8563

$q$

Test charge

$C$

8746

Relation

Development

ID:(11596, 0)