A steady current flows across the plane boundary between two conductors of conductivities \(\sigma_{1}\) and \(\sigma_{2}\). Find a relation between the angles \(\theta_{1}\) and \(\theta_{2}\) which the current flow makes with the normal to the boundary on the two sides. What will be the effect if one conductor is a metal and the other an electrolyte?

Electrical conductivity is a measure of a material's ability to conduct electric current. It is denoted by the symbol \(\sigma\) and is defined as the ratio of the current density \(\vec{J}\) to the electric field \(\vec{E}\) in a material: \[ \sigma = \frac{\vec{J}}{\vec{E}} \. \] The higher the conductivity, the better a material is at allowing electric charge to flow through it. Metals typically have high electrical conductivity, while insulators have low conductivity. In the exercise provided, the conductivities of two different conductors, \(\sigma_{1}\) and \(\sigma_{2}\), play a crucial role in determining the current flow directions on either side of a boundary.

Electrical conductivity is not only a property of the material but can be influenced by factors such as temperature, impurities in the material, and the physical structure of the material. When a steady current flows across a boundary between two conductors with different conductivities, the behavior of the current at that boundary is governed by the principles of continuity and the relationship between the electric field and the conductivities.

Snell's Law for current flow is analogous to Snell's Law in optics, which relates the angles of incidence and refraction of light at the boundary between two different media. For electric current, we can derive a similar relationship by considering the current densities and the angles they make with the normal to the boundary. In this context, Snell’s Law states that the ratio of the sine of the angle of the current flow in the first conductor to the sine of the angle of the current flow in the second conductor is equal to the inverse ratio of their conductivities: \[ \frac{\sin(\theta_1)}{\sin(\theta_2)} = \frac{\sigma_2}{\sigma_1} \. \] This formula helps in determining how the direction of current flow changes when it passes from one conductor to another with a different conductivity.

It's important to understand that the current will bend towards the normal when moving to a region of lower conductivity, and away from the normal when moving to a region of higher conductivity. This principle is crucial in applications such as electrical engineering and materials science, where controlling the direction of current flow is essential.

Current density is a vector quantity that represents the amount of electric current flowing per unit area of a cross-section, perpendicular to the direction of flow. It is given by the equation: \[ \vec{J} = \sigma \vec{E} \. \] Where \(\vec{J}\) is the current density, \(\sigma\) is the electrical conductivity, and \(\vec{E}\) is the electric field. Current density is significant because it describes not only how much current is flowing but also the distribution of that current across a given area.

In the context of the exercise, ensuring continuity of current density across the plane boundary between two conductors means that despite differences in conductivity, the total current must remain the same. Therefore, if one conductor is more conductive than the other, the current will spread out more in the former to ensure that the current per unit area remains constant.

Electric field continuity is a fundamental concept in electromagnetism, especially at the interface between two different conductive materials. At the boundary, the tangential component of the electric field must be continuous. This means that despite variations in material properties like conductivity, the horizontal component of the electric field remains the same across the boundary.

For current flow across a boundary, applying the idea of electric field continuity is important, as seen in the exercise, where we equate the product of conductivity and the electric field for both conductors, factoring in the angle to the normal: \[ \sigma_1E_1 \cos(\theta_1) = \sigma_2E_2 \cos(\theta_2) \. \] This relationship assures that at the point of transition from one material to the other, despite any changes in conductivity, the current does not experience any 'breaks' or sudden changes in flow, resulting in a smooth transition of power across materials.

Metals and electrolytes exhibit very different electrical conductances due to their physical and chemical makeup. Metals have free electrons that move easily under the influence of an electric field, resulting in high electrical conductivity. In contrast, electrolytes conduct electricity through the movement of ions, which is typically less efficient, resulting in lower conductivity.

In the exercise's context, when a current passes from a metal to an electrolyte, the sharp difference in conductance affects the current flow's direction substantially. The current spreads out within the metal and is more narrowly focused within the electrolyte, as explained by the large difference in angles \(\theta_1\) and \(\theta_2\) concluded from the Snell's Law for current. This behavior can be important in designing electrochemical cells, batteries, and corrosion prevention systems, where metal-electrolyte interfaces are common.