The solid metal sphere of radius a=0.200 m shown in the figure has a surface charge distributio

step 1 Hi there, so for this problem, the solid metal sphere of radius A equals to 0 .2 meters shown in

The solid metal sphere of radius a=0.200 m shown in the...

The solid metal sphere of radius $a=0.200 \mathrm{~m}$ shown in the figure has a surface charge distribution of $\sigma$. The potential difference between the surface of the sphere and a point $P$ at a distance $r_{\mathrm{p}}=0.500 \mathrm{~m}$ from the center of the sphere is $\Delta V=V_{\text {surfhee }}-V_{\mathrm{p}}=+4 \pi \mathrm{V}=$ $+12.566 \mathrm{~V}$. Determine the value of $\sigma$.

The solid metal sphere of radius $a=0.200 \mathrm{~m}$ shown in the figure has a surface charge distribution of $\sigma$. The potential difference between the surface of the sphere and a point $P$ at a distance $r_{\mathrm{p}}=0.500 \mathrm{~m}$ from the center of the sphere is $\Delta V=V_{\text {surfhee }}-V_{\mathrm{p}}=+4 \pi \mathrm{V}=$
$+12.566 \mathrm{~V}$. Determine the value of $\sigma$.

Key Concepts

Relationship Between Surface Charge, Total Charge, and Potential

The connection between the surface charge density (?), the total charge (Q), and the electric potential is critical in solving electrostatic problems. The total charge on the surface of a sphere is obtained by multiplying ? by the surface area of the sphere. This total charge then creates an electric potential at points in the surrounding space, which can be calculated using the formula for the potential of a point charge. Understanding this relationship allows one to determine the unknown ? when provided with the potential difference.

Surface Charge Density

Surface charge density, denoted ?, represents the amount of electric charge per unit area on the surface of a conductor. It is a fundamental quantity in electrostatics that relates the total charge on an object to its geometrical properties. Using ?, one can easily compute the net charge by multiplying it by the total surface area of the charged body.

Electric Potential of a Charged Sphere

For a spherical conductor, the electric potential at a point outside the sphere is equivalent to that produced by a point charge located at the center. This simplification is a direct consequence of the spherical symmetry and is useful for calculating potential differences. The potential at any point outside the sphere varies inversely with the distance from the center, which facilitates the determination of ? when combined with known potential differences.

Gauss's Law

Gauss's Law is a fundamental principle in electrostatics that relates the electric flux through a closed surface to the charge enclosed by that surface. For objects with high symmetry, such as spheres, Gauss's Law simplifies the computation of the electric field. Once the electric field is determined, it can be integrated to find potentials, linking the distribution of surface charge to measurable potential differences.

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