Two spherical conducting wires A and B are connected to the same potential difference. Wire A is three times as long as wire B, with a radius double that of wire B and resistivity doubles that of wire B. What is the ratio power delivered to wire A to the power delivered to wire B? P A/P B
given answers
a. 1/3
b. 2/3
c. 3/3
d. 3/2
e. 4/2
someone please help, I cant figure out the ratio.
To find the ratio of the power delivered to wire A (PA) to the power delivered to wire B (PB), we need to consider the factors that affect power in a circuit.
The power (P) delivered to a wire can be calculated using the formula:
P = (V^2) / R,
where V is the potential difference across the wire and R is its resistance.
Given that wires A and B are connected to the same potential difference, the ratio of their powers can be expressed as:
PA / PB = (VA^2 / RA) / (VB^2 / RB),
where VA and VB are the potential differences across wires A and B, respectively, and RA and RB are their resistances.
Let's break down the problem step by step:
Step 1: Compare the lengths of the wires.
It is given that wire A is three times as long as wire B. This means that the length ratio LA / LB = 3.
Step 2: Compare the radii of the wires.
Wire A has a radius double that of wire B. This means that the radius ratio rA / rB = 2.
Step 3: Compare the resistivities of the wires.
The resistivity of wire A is double that of wire B.
Now, let's tackle each component separately:
Resistance:
The resistance of a wire is given by the formula:
R = (ρ * L) / A,
where ρ is the resistivity of the material, L is the length of the wire, and A is the cross-sectional area of the wire.
Since wire A has a resistivity that is double that of wire B, we can write:
RA / RB = (ρA * LA) / (ρB * LB).
Length:
We know that the length ratio of the wires is LA / LB = 3.
Radius:
The radius ratio of the wires is rA / rB = 2.
Now, substituting these values into the expression for resistance ratio:
RA / RB = (ρA * LA) / (ρB * LB) = (2 * LA) / (LB * 3).
Simplifying this expression, we get:
RA / RB = 2LA / (3LB).
Finally, let's substitute this resistance ratio into the power ratio formula:
PA / PB = (VA^2 / RA) / (VB^2 / RB).
Since the potential differences across the wires are the same, we can simplify the above formula to:
PA / PB = (VA^2 / RA) / (VB^2 / RB) = (VA^2 * RB) / (VB^2 * RA).
Now, substitute the resistance ratio we found earlier:
PA / PB = (VA^2 * RB) / (VB^2 * RA) = (VA^2 * RB) / (VB^2 * (2LA / 3LB)).
Simplifying further:
PA / PB = (VA^2 * RB * 3LB) / (VB^2 * (2LA)).
Finally, we can simplify the equation to:
PA / PB = (3RB * LB) / (2LA).
From the given values, we can see that this ratio is equal to 2/3 (option b). Therefore, the correct answer is (b) 2/3.