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Plahaz 2ɛ0[h²+a²]³/² for the ring below (radius of a, at h), A) calculate the E (at h) of a disk with the radius of a. B) If a →∞o, what will be the E of the disk at the height of h and height of ∞o? Help: P₁ = pędr, ar If E= X dE dEp dE₂ α h R a dl

Plahaz 2ɛ0[h²+a²]³/² for the ring below (radius of a, at h), A) calculate the E (at h) of a disk with the radius of a. B) If a →∞o, what will be the E of the disk at the height of h and height of ∞o? Help: P₁ = pędr, ar If E= X dE dEp dE₂ α h R a dl

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Please provide a detailed step-by-step solution for the problem shown in the attached image. Thank you!
### Electric Field Calculation for a Ring **If** \[ E = \frac{\rho_l a h a_z}{2 \varepsilon_0 [h^2 + a^2]^{3/2}} \] for the ring below (radius of \( a \), at \( h \)), **Tasks:** **A)** Calculate the \( E \) (at \( h \)) of a disk with the radius of \( a \). **B)** If \( a \rightarrow \infty \), what will be the \( E \) of the disk at the height of \( h \) and height of \(\infty\)? **Help:** \[ \rho_l = \rho_s dr, \quad a \rightarrow r \] #### Diagram Explanation: The diagram shows a 3D coordinate system with a horizontal circular ring positioned within the xy-plane. The following elements are depicted: - The **z-axis** is vertical and points upwards through the center of the ring. - The **radius** \( a \) of the ring is shown in the xy-plane. - A vector **\( R \)** points from a point on the ring to a point directly above or below on the z-axis. - **\( h \)** is the height from the center of the ring to the point on the z-axis above it. - The differential length element \( dl \) is indicated along the perimeter of the ring. - The vector components \( dE \), \( dE_z \), and \( dE_\rho \) are shown: - \( dE \) represents the differential electric field vector at a point. - \( dE_z \) is the z-component of \( dE \). - \( dE_\rho \) is the radial component of \( dE \). - \( \alpha \) is the angle between \( dE \) and the components, indicating their projections on the z and radial directions. This setup is used to derive or calculate the electric field at different heights from the ring or disk, based on symmetries and integration over differential elements.
Transcribed Image Text:### Electric Field Calculation for a Ring **If** \[ E = \frac{\rho_l a h a_z}{2 \varepsilon_0 [h^2 + a^2]^{3/2}} \] for the ring below (radius of \( a \), at \( h \)), **Tasks:** **A)** Calculate the \( E \) (at \( h \)) of a disk with the radius of \( a \). **B)** If \( a \rightarrow \infty \), what will be the \( E \) of the disk at the height of \( h \) and height of \(\infty\)? **Help:** \[ \rho_l = \rho_s dr, \quad a \rightarrow r \] #### Diagram Explanation: The diagram shows a 3D coordinate system with a horizontal circular ring positioned within the xy-plane. The following elements are depicted: - The **z-axis** is vertical and points upwards through the center of the ring. - The **radius** \( a \) of the ring is shown in the xy-plane. - A vector **\( R \)** points from a point on the ring to a point directly above or below on the z-axis. - **\( h \)** is the height from the center of the ring to the point on the z-axis above it. - The differential length element \( dl \) is indicated along the perimeter of the ring. - The vector components \( dE \), \( dE_z \), and \( dE_\rho \) are shown: - \( dE \) represents the differential electric field vector at a point. - \( dE_z \) is the z-component of \( dE \). - \( dE_\rho \) is the radial component of \( dE \). - \( \alpha \) is the angle between \( dE \) and the components, indicating their projections on the z and radial directions. This setup is used to derive or calculate the electric field at different heights from the ring or disk, based on symmetries and integration over differential elements.
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