Question:
What is difference between an open surface and a closed surface ? Draw elementary surface vector \(d\vec{S}\) for a spherical surface S.
What is difference between an open surface and a closed surface ? Draw elementary surface vector \(d\vec{S}\) for a spherical surface S.
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A closed surface has no boundary and encloses a volume. For a closed surface \(d\vec{S}\) points along the outward normal.
Updated On: Oct 27, 2026
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Solution and Explanation
Step 1: Understanding the Concept:
Surfaces are used in electrostatics to count field lines (flux). To do this we must know whether the surface encloses a volume and in which direction we measure its area.
Step 2: Open surface and closed surface:
An open surface has a boundary edge (rim). It does not enclose any volume. We can go from one side to the other without crossing the surface, by walking around the rim. Examples: a flat sheet, a circular disc, a hemispherical bowl.
A closed surface has no boundary edge. It divides the space into an inside and an outside. To go from inside to outside we must cross the surface. Examples: a sphere, a cube, a cylinder with both end caps.
Step 3: Elementary surface vector for a sphere:
The elementary surface vector is \(d\vec{S} = dS\,\hat{n}\). Its magnitude \(dS\) is the area of a very small patch. Its direction is along the normal to that patch. For a closed surface the normal is taken outward.
For a spherical surface S with centre O, take a small patch at point P. The outward normal at P is along the radius OP, so \(d\vec{S}\) points radially outward, away from the centre.
Diagram in words: draw a circle for the sphere, mark a small patch on it, and draw an arrow from the patch straight outward along the radius, labelled \(d\vec{S}\). This arrow is perpendicular to the patch.
Final Answer:
An open surface has a rim and encloses no volume. A closed surface has no rim and encloses a volume. For a sphere, \(d\vec{S}\) is drawn radially outward (outward normal) on each small patch.
Surfaces are used in electrostatics to count field lines (flux). To do this we must know whether the surface encloses a volume and in which direction we measure its area.
Step 2: Open surface and closed surface:
An open surface has a boundary edge (rim). It does not enclose any volume. We can go from one side to the other without crossing the surface, by walking around the rim. Examples: a flat sheet, a circular disc, a hemispherical bowl.
A closed surface has no boundary edge. It divides the space into an inside and an outside. To go from inside to outside we must cross the surface. Examples: a sphere, a cube, a cylinder with both end caps.
Step 3: Elementary surface vector for a sphere:
The elementary surface vector is \(d\vec{S} = dS\,\hat{n}\). Its magnitude \(dS\) is the area of a very small patch. Its direction is along the normal to that patch. For a closed surface the normal is taken outward.
For a spherical surface S with centre O, take a small patch at point P. The outward normal at P is along the radius OP, so \(d\vec{S}\) points radially outward, away from the centre.
Diagram in words: draw a circle for the sphere, mark a small patch on it, and draw an arrow from the patch straight outward along the radius, labelled \(d\vec{S}\). This arrow is perpendicular to the patch.
Final Answer:
An open surface has a rim and encloses no volume. A closed surface has no rim and encloses a volume. For a sphere, \(d\vec{S}\) is drawn radially outward (outward normal) on each small patch.
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