Spherical Charge System with Zero External Field: What spherical charge distribution creates the field given at right, wherer is defined as in spherical coordinates? Note the discontinuity in the field: what does this require at the surface? Po rr rs. E(F)= 380 0 r>.
Spherical Charge System with Zero External Field: What spherical charge distribution creates the field given at right, wherer is defined as in spherical coordinates? Note the discontinuity in the field: what does this require at the surface? Po rr rs. E(F)= 380 0 r>.
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![### Spherical Charge System with Zero External Field
**Problem Statement:**
Consider a spherical charge distribution that results in the electric field described below. The goal is to determine the nature of this charge distribution, given that \( r \) is defined in spherical coordinates. Additionally, examine the discontinuity in the field and discuss its implications at the surface.
**Electric Field Description:**
The electric field \( \mathbf{E}(r) \) is defined as:
\[
\mathbf{E}(r) =
\begin{cases}
\frac{\rho_0}{3\epsilon_0}r \, \mathbf{\hat{r}} & r \leq a \\
0 & r > a
\end{cases}
\]
**Explanation:**
- For \( r \leq a \): The electric field inside the sphere is radial and its magnitude increases linearly with \( r \), scaled by the factor \(\frac{\rho_0}{3\epsilon_0}\).
- For \( r > a \): The electric field outside the sphere is zero, indicating no external influence.
**Discussion:**
- The discontinuity in the electric field at \( r = a \) suggests a boundary condition that must be satisfied by the charge distribution. Understanding this requires analyzing the behavior of the field and its relation to the charge density within the spherical region.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F099a9854-7cc3-46b9-a37c-ec0486b4a12a%2Ff096a633-480e-4b79-a378-b1e33f207975%2Fosiynpr_processed.png&w=3840&q=75)
Transcribed Image Text:### Spherical Charge System with Zero External Field
**Problem Statement:**
Consider a spherical charge distribution that results in the electric field described below. The goal is to determine the nature of this charge distribution, given that \( r \) is defined in spherical coordinates. Additionally, examine the discontinuity in the field and discuss its implications at the surface.
**Electric Field Description:**
The electric field \( \mathbf{E}(r) \) is defined as:
\[
\mathbf{E}(r) =
\begin{cases}
\frac{\rho_0}{3\epsilon_0}r \, \mathbf{\hat{r}} & r \leq a \\
0 & r > a
\end{cases}
\]
**Explanation:**
- For \( r \leq a \): The electric field inside the sphere is radial and its magnitude increases linearly with \( r \), scaled by the factor \(\frac{\rho_0}{3\epsilon_0}\).
- For \( r > a \): The electric field outside the sphere is zero, indicating no external influence.
**Discussion:**
- The discontinuity in the electric field at \( r = a \) suggests a boundary condition that must be satisfied by the charge distribution. Understanding this requires analyzing the behavior of the field and its relation to the charge density within the spherical region.
Expert Solution
Step 1
Solution:
Let us consider a uniformly charged non conducting sphere with a charged cavity displaced at a distance r from the center of the sphere. Let a be the radius of the sphere and rho be the charge density of the sphere. The cavity is filled with equal and opposite charge of the sphere.
Now electric field inside the sphere can be written as,
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