A hemisphere of radius R is placed in a charge-free region...

A hemisphere of radius $R$ is placed in a charge-free region of space where a uniform electric field exists of magnitude $E$ directed perpendicular to the hemisphere's circular base (Fig. $22-50$ ). (a) Using the definition of $\Phi_{E}$ through an "open" surface, calculate (via explicit integration) the electric flux through the hemisphere. [Hint: In Fig. $22-50$ you can see that, on the surface of a sphere, the infinitesimal area located between the angles $\theta$ and $\theta+d \theta$ is $\left.d A=(2 \pi R \sin \theta)(R d \theta)=2 \pi R^{2} \sin \theta d \theta .\right] \quad(b) \quad$ Choose $\quad$ an appropriate gaussian surface and use Gauss's law to much more easily obtain the same result $d A=2 \pi R^{2} \sin \theta d \theta$ for the electric flux through the hemisphere.

A hemisphere of radius $R$ is placed in a charge-free region of space where a uniform electric field exists of magnitude $E$ directed perpendicular to the hemisphere's circular base (Fig. $22-50$ ). (a) Using the definition of $\Phi_{E}$ through an "open" surface, calculate (via explicit integration) the electric flux through the hemisphere. [Hint: In Fig. $22-50$ you can see that, on the surface of a sphere, the infinitesimal area located between the angles $\theta$ and $\theta+d \theta$ is $\left.d A=(2 \pi R \sin \theta)(R d \theta)=2 \pi R^{2} \sin \theta d \theta .\right] \quad(b) \quad$ Choose $\quad$ an
appropriate gaussian surface and use Gauss's law to much more easily obtain the same result $d A=2 \pi R^{2} \sin \theta d \theta$
for the electric flux through the hemisphere.

Key Concepts

Electric Flux

Electric flux is the measure of the amount of electric field passing through a given surface. It is defined mathematically as the surface integral of the electric field over that surface, represented as the dot product of the electric field vector and the differential area vector. This concept is central in electromagnetism as it quantifies how fields interact with surfaces, and forms the cornerstone of Gauss's law.

Surface Integration in Spherical Coordinates

When calculating quantities over curved surfaces, such as a hemisphere or a sphere, it is often convenient to use spherical coordinates. In this system, the differential area element on a spherical surface is expressed in terms of angular variables, typically involving a factor that accounts for the sine of one of the angles. This formulation simplifies the integration over a spherical surface by reducing a two-dimensional area integration into a single integral over an angle, with the geometry encapsulated in the differential element.

Gauss's Law

Gauss's law relates the net electric flux through a closed surface to the charge enclosed by that surface. It is a powerful tool in electrostatics, especially when applied to problems with high symmetry. By choosing an appropriate Gaussian surface, the problem of calculating electric flux can be greatly simplified, as the symmetry often allows for a straightforward evaluation of the surface integral, even when the actual distribution of the electric field is complex.

Gaussian Surface Selection

Choosing an appropriate Gaussian surface is crucial when applying Gauss's law. The key is to select a surface that exploits the Symmetry of the problem so that the electric field is either constant over portions of the surface or orthogonal/parallel to the surface elements. This deliberate choice simplifies the calculation of flux, turning a potentially challenging surface integral into a more manageable problem, and directly relates the geometry of the chosen surface to the physical charges enclosed or the external field configuration.

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