### Electric Potential and Field due to a Conducting Sphere in an External Electric FieldWhen an uncharged conducting sphere of radius \( a \) is placed at the origin of an xyz coordinate system that lies in an initially uniform electric field \(\vec{E} = E_0 \hat{k}\), the resulting electric potential is given by:\[ V(x, y, z) = V_0 \]for the points inside the sphere, and\[ V(x, y, z) = V_0 - E_0 z + \frac{E_0 a^3 z}{(x^2 + y^2 + z^2)^{3/2}} \]for points outside the sphere, where \( V_0 \) is the (constant) electric potential on the conductor.### ObjectiveUse the above equations to determine the \( x \), \( y \), and \( z \) components of the resulting electric field in the following regions. Use the following notations as necessary: \( x \), \( y \), \( z \), \( a \), and \( E_0 \).**(a) Inside the Sphere**\[ E_x = \] (Input Field)\[ E_y = \] (Input Field)\[ E_z = \] (Input Field)**(b) Outside the Sphere**\[ E_x = \] (Input Field)\[ E_y = \] (Input Field)\[ E_z = \] (Input Field)### GuidanceFor further assistance, please refer to additional resources by clicking on the "Need Help? Read It" button.### Explanation of Diagrams/GraphsThe image does not contain any graphs or diagrams. The key mathematical relationships provided define the electric potential inside and outside the conducting sphere influenced by an external uniform electric field. Calculation of the electric field components will involve differentiating the potential functions with respect to \( x \), \( y \), and \( z \).

Answered: Image /qna-images/answer/de714881-dc79-4edf-bcd5-f4440df7a7c0.jpg

When an uncharged conducting sphere of radius a is placed at the origin of an xyz coordinate system that lies in an initially uniform electric fieldE = E, k, the resulting electric potential is V (x,y,z) = Vo for the points inside the sphere and V (x,y,z) = Vo – Egz + Ega³z (x² + y² + z²)3/2 for points outside the sphere, where V is the (constant) electric potential on the conductor. Use this equation to determine the x, y, and z components of the resulting electric field in the following regions. (Use the following as necessary: X, y, z, a, and Eo:)

When an uncharged conducting sphere of radius a is placed at the origin of an xyz coordinate system that lies in an initially uniform electric fieldE = E, k, the resulting electric potential is V (x,y,z) = Vo for the points inside the sphere and V (x,y,z) = Vo – Egz + Ega³z (x² + y² + z²)3/2 for points outside the sphere, where V is the (constant) electric potential on the conductor. Use this equation to determine the x, y, and z components of the resulting electric field in the following regions. (Use the following as necessary: X, y, z, a, and Eo:)

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Electric Charges and Fields

One of the fundamental properties is the electromagnetic property. Charge cannot be destroyed by any process and this contributes formally to the law of charge conservation.

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### Electric Potential and Field due to a Conducting Sphere in an External Electric Field

When an uncharged conducting sphere of radius \( a \) is placed at the origin of an xyz coordinate system that lies in an initially uniform electric field \(\vec{E} = E_0 \hat{k}\), the resulting electric potential is given by:

\[ V(x, y, z) = V_0 \]

for the points inside the sphere, and

\[ V(x, y, z) = V_0 - E_0 z + \frac{E_0 a^3 z}{(x^2 + y^2 + z^2)^{3/2}} \]

for points outside the sphere, where \( V_0 \) is the (constant) electric potential on the conductor.

### Objective
Use the above equations to determine the \( x \), \( y \), and \( z \) components of the resulting electric field in the following regions. Use the following notations as necessary: \( x \), \( y \), \( z \), \( a \), and \( E_0 \).

**(a) Inside the Sphere**

\[ E_x = \] (Input Field)
\[ E_y = \] (Input Field)
\[ E_z = \] (Input Field)

**(b) Outside the Sphere**

\[ E_x = \] (Input Field)
\[ E_y = \] (Input Field)
\[ E_z = \] (Input Field)

### Guidance
For further assistance, please refer to additional resources by clicking on the "Need Help? Read It" button.

### Explanation of Diagrams/Graphs
The image does not contain any graphs or diagrams. The key mathematical relationships provided define the electric potential inside and outside the conducting sphere influenced by an external uniform electric field. Calculation of the electric field components will involve differentiating the potential functions with respect to \( x \), \( y \), and \( z \).

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