(a) Compute the electrical conductivity of a \(7.0-\mathrm{mm}(0.28 \text { -in. })\) diameter cylindrical sil icon specimen \(57 \mathrm{mm}(2.25\) in.) long in which a current of 0.25 A passes in an axial direction. A voltage of \(24 \mathrm{V}\) is measured across two probes that are separated by \(45 \mathrm{mm}(1.75 \text { in. })\) (b) Compute the resistance over the entire \(57 \mathrm{mm}(2.25 \text { in. })\) of the specimen.

At the heart of electronics and electrical engineering is a fundamental principle known as Ohm's Law. It is the backbone of our understanding of how electric currents behave in materials. Simply put, Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance between them. The mathematical expression of Ohm's Law is as follows:

\[ I = \frac{V}{R} \]

where \(I\) is the current measured in amperes (A), \(V\) is the voltage measured in volts (V), and \(R\) is the resistance measured in ohms (Ω). This relationship shows us that if you increase the voltage across a conductor, more current will flow provided the resistance remains constant. Conversely, if you increase the resistance, less current flows for a constant voltage. The concept is essential for troubleshooting circuits and designing electronic systems.

When dealing with materials, especially in the context of conducting electricity, we refer to a property known as resistivity. It is a measure of how strongly a material opposes the flow of electric current.

Resistivity (\rho) is calculated by using the formula:

\[ \rho = R \times \frac{A}{L} \]

Here, \(R\) denotes resistance in ohms, \(A\) is the cross-sectional area through which the current flows, and \(L\) is the length over which we are measuring the resistance. The unit of resistivity is ohm-meters (Ω·m) or in context-specific units like ohm-millimeters squared per meter (\( Ω\cdot mm^2/m \)). Understanding resistivity is crucial for material selection, especially in the design of wires and components. A lower resistivity implies that the material is a good conductor, while a high resistivity indicates poor conductivity. This concept helps us understand why materials like copper are widely used in electrical wiring and why insulators like rubber are used to prevent unwanted current flow.

To understand how electric current gets affected as it passes through an object, we calculate the object's resistance. The calculation involves taking into account both the geometry and the material of the object. The formula to calculate the resistance (R) of a cylindrical object, like a wire, is derived from its resistivity (\(ρ\)). We use the formula:

\[ R = \rho \times \frac{L}{A} \]

where \(L\) is the length of the object through which the current flows, and \(A\) is its cross-sectional area. It is worth mentioning that when calculating the resistance for cylindrical shapes, the area (A) is determined by the radius (\(r\)) or diameter (\(d\)) of the cylinder using the area formula for circles. An increased length of the conductor or decreased area will result in higher resistance.

• Length: Longer conductors have more resistance because electrons have a longer distance to travel, colliding more frequently with the atoms in the material.
• Area: Thinner wires have less area for current to flow through, which increases resistance.

Understanding this relationship enables us to optimize electronic circuits for desired behaviors by adjusting wire lengths and thicknesses. It's particularly crucial when considering the efficiency of power transmission in our electrical grids.