Question

How Does The Resistance Of Wire Depend On Its Radius?

Answer 1

An object's electrical resistance is a measure of its resistance to the flow of electric current. Electrical conductance, which measures the ease with which an electric current flows, is its reciprocal quantity. Electrical resistance and mechanical friction have some conceptual similarities. The ohm ($\Omega$) is the SI unit for electrical resistance.

Complete answer:
The resistance of an object is largely determined by two factors: the material used and the form of the thing. The resistance of a substance is inversely related to its cross-sectional area; for example, a thick copper wire has lower resistance than a thin copper wire that is otherwise equal. Also, the resistance of a particular material is related to its length; for example, a long copper wire has higher resistance than a short copper wire that is otherwise equal. As a result, the resistance R and conductance G of a homogeneous cross-section cable may be calculated as
$R = \rho \dfrac{\ell }{A}$
We already know that a wire's resistance is inversely proportional to its cross-sectional area. The cross-sectional area of the wire, on the other hand, is proportional to the square of the wire's radius. This indicates that the wire's resistance and radius are inversely proportional to one another. As a result, the resistance of wire reduces as the radius rises. As a result, the resistance of a wire is proportional to its radius.
where l is the conductor's length in metres (m), A is the conductor's cross-sectional area in square metres (${m^2}$), $\sigma$(sigma) is the electrical conductivity in Siemens per metre ($S{m^{ - 1}}$), and $\rho$(rho) is the material's electrical resistivity (also known as specific electrical resistance) in ohm-metres ($\Omega$m). Because resistivity and conductivity are proportionality constants, they are determined only by the wire's substance, not its geometry. The reciprocal of resistivity and conductivity is: $\rho = \dfrac{1}{\sigma }$ The capacity of a substance to resist electric current is measured by its resistivity.

Note:
When dealing with alternating current (AC), the skin effect slows current flow towards the centre of the conductor, thus this formula isn't perfect. As a result, the geometrical cross-section differs from the effective cross-section through which current flows, resulting in greater resistance than intended. Similarly, the proximity effect causes the resistances of two conductors carrying AC current to rise. Substantial conductors carrying big currents, such as busbars in an electrical substation, or large power cables carrying more than a few hundred amperes, exhibit these effects at commercial power frequency.