How to determine equivalent resistance

Question

Let us be two cylindrical conductors connected in parallel, to which a potential difference of V = 170V is applied. The two conductors are made of the same material, but the first is 6 times the length of the second, and  the radius of the second. The resistance of the second is R2 = 469Ω. Determine the equivalent resistance.

Answer 1

The resistance of a cylindrical conductor is given by the formula:

R=ρ*L/A

where:

( R ) is the resistance,

( \rho ) is the resistivity of the material,

( L ) is the length of the conductor, and

( A ) is the cross-sectional area of the conductor.

Given that the two conductors are made of the same material and are connected in parallel, the equivalent resistance (( R_{eq} )) can be calculated using the formula for parallel resistors:

1���=1�1+1�2Req​1​=R1​1​+R2​1​

We know that ( R_2 = 469 , \Omega ), and that the first conductor is 6 times the length of the second, and has half the radius. Therefore, the resistance of the first conductor (( R_1 )) can be calculated as:

�1=�6�14��2=24�2R1​=ρ41​πd26L​=24R2​

Substituting ( R_1 = 24 R_2 ) into the formula for ( R_{eq} ), we get:

1���=124�2+1�2=2524�2Req​1​=24R2​1​+R2​1​=24R2​25​

Solving for ( R_{eq} ), we find:

���=24�225Req​=2524R2​​

Substituting ( R_2 = 469 , \Omega ) into the equation, we can calculate the equivalent resistance.