Question

ПОЛЬЗУЯСЬ ТЕОРЕМОЙ ГАУССА,НАЙТИ НАПРЯЖЕННОСТЬ ЭЛЕКТРИЧЕСКОГО ПОЛЯ ВНУТРИ ОДНОРОДНО ЗАРЯЖЕННОГО ШАРА РАДИУСАR.ОБЪЕМНАЯ ПЛОТНОСТЬЗАРЯДА p

Answer 1

Воспользуемся теоремой Гаусса:

$\int{E}\, dS=\sum_{i=1}^{n}q_i$

Так напряженность внутри сферы постоянная, получаем:

$\int{E}\, dS=E\int{}\, dS=ES$

Объемная плотность заряда равна:

$p=\frac{Q}{V}$

$Q=pV$

$V=\frac{SR}{3}$

Получаем:

$ES=Q$

$ES=pV$

$ES=p\frac{SR}{3}$

$E=\frac{pR}{3}$

Ответ: $E=\frac{pR}{3}$

Answer 2

Finding the Electric Field Inside a Uniformly Charged Sphere Using Gauss's Law

To find the electric field inside a uniformly charged sphere using Gauss's Law, we can apply the following steps:

1. Define the problem: We have a uniformly charged sphere with a radius R and a volume charge density p.

2. Choose a Gaussian surface: We need to choose a Gaussian surface that encloses the region where we want to find the electric field. In this case, we can choose a spherical Gaussian surface with a radius r less than or equal to R.

3. Apply Gauss's Law: Gauss's Law states that the electric flux through a closed surface is equal to the total charge enclosed divided by the permittivity of free space (ε₀). Mathematically, it can be expressed as:

Φ = Q / ε₀

where Φ is the electric flux, Q is the total charge enclosed, and ε₀ is the permittivity of free space.

4. Calculate the electric flux: Since the electric field is radially symmetric, the electric flux through the Gaussian surface will be constant and equal to the electric field magnitude multiplied by the surface area of the Gaussian sphere. Mathematically, it can be expressed as:

Φ = E * 4πr²

where E is the magnitude of the electric field and r is the radius of the Gaussian sphere.

5. Calculate the total charge enclosed: The total charge enclosed within the Gaussian surface is equal to the volume charge density multiplied by the volume of the sphere. Mathematically, it can be expressed as:

Q = p * (4/3)πR³

where p is the volume charge density and R is the radius of the sphere.

6. Equate the electric flux and the total charge enclosed: By equating the electric flux and the total charge enclosed, we can solve for the magnitude of the electric field inside the sphere. Mathematically, it can be expressed as:

E * 4πr² = p * (4/3)πR³ / ε₀

Simplifying the equation, we get:

E = p * R³ / (3ε₀r²)

where E is the magnitude of the electric field, p is the volume charge density, R is the radius of the sphere, ε₀ is the permittivity of free space, and r is the radius of the Gaussian sphere.

Therefore, the magnitude of the electric field inside a uniformly charged sphere can be calculated using the formula:

E = p * R³ / (3ε₀r²)

Please note that this formula assumes that the charge distribution is spherically symmetric and that the Gaussian surface is chosen appropriately.

Let me know if you need any further assistance!