To calculate the ratio of the potential difference (p.d.) across wire A to that across wire B, we need to consider the relationship between resistance, length, and resistivity. The resistance of a wire is given by the formula:
R = (ρ * L) / A
where R is the resistance, ρ is the resistivity, L is the length, and A is the cross-sectional area.
Given that wire A is 0.5m long with a resistivity of 7.0×10^-5, and wire B is 1.0m long with a resistivity of 3.5×10^-5, we can calculate the ratio of their resistances.
For wire A:
Resistance_A = (7.0×10^-5 * 0.5) / A
For wire B:
Resistance_B = (3.5×10^-5 * 1.0) / A
Since both wires have the same thickness, their cross-sectional areas (A) are equal and cancel out when calculating the ratio.
Now, let's assume that the current flowing through both wires is the same. According to Ohm's Law, the potential difference (V) across a resistor is directly proportional to the resistance:
V = I * R
where V is the potential difference, I is the current, and R is the resistance.
Therefore, the ratio of the potential differences across wire A (V_A) to wire B (V_B) will be the same as the ratio of their resistances:
V_A / V_B = Resistance_A / Resistance_B
Substituting the expressions for Resistance_A and Resistance_B:
V_A / V_B = [(7.0×10^-5 * 0.5) / A] / [(3.5×10^-5 * 1.0) / A]
Simplifying:
V_A / V_B = (7.0×10^-5 * 0.5) / (3.5×10^-5 * 1.0)
V_A / V_B = 0.5 / 1.0
V_A / V_B = 0.5
Therefore, the ratio of the potential difference across wire A to that across wire B is 0.5.