Ch 9 Model Questions

Model Questions

Ch 9 Current and Resistance

Concept Questions
Long Answer Questions

Concept Questions

c1. Rank currents in wires

All the wires in the figure are made out of the same material and have the same diameter. Rank in order, from largest to smallest, the currents $I_1$ to $I_4$.

Current conserved, so $I_1 = I_4$.
Current splits in two at junction, so $I_1 > I_2$.
Wires carrying $I_1$ to $I_4$ have the same resistance, so $I_2 = I_3$.

$I_1 = I_4 > I_2 = I_3$

c2. Current and wire diameter

Wire 1 and wire 2 are made from the same metal. Wire 2 has a larger diameter than wire 1. The electric field strengths $E_1$ and $E_2$ in the wires are equal.

For the following comparisons of wires 1 and 2, state whether they are equal, or which is greater.

Compare the conductivites, $\sigma_1$ and $\sigma_2$.
Compare the current densities, $J_1$ and $J_2$.
Compare the currents, $I_1$ and $I_2$.
Compare the values of the electron drift speeds, $(v_d)_1$ and $(v_d)_2$.

\(\sigma_2 = \sigma_1\); wires have
the same metal
\(J_1 = J_2\); because \(J=\sigma E\)
\(I_1 \lt I_2\); because \(I = J A\) and \(A_1 \lt A_2\)
\((v_d)_1 = (v_d)_2\); \(J = n_e e v_d\), and number density \(n_e\) is the same

c3. Wire with variable conductivity

A wire consists of two equal-diameter segments. Their conductivities differ, with $\sigma_2 > \sigma_1$. The current in segment 1 is $I_1$.

Compare the values of the currents in the two segments, $I_1$ and $I_2$.
Compare the current densities, $J_1$ and $J_2$.
Compare the strengths of the electric fields, $E_1$ and $E_2$.

\(I_1 = I_2\)
\(J_1 = J_2\)
\(E_1 > E_2\)

c4. Current comparison

The two circuits use identical batteries and wires of equal diameters. Rank in order, from largest to smallest, the currents $I_1$, $I_2$, $I_3$ and $I_4$.

Current conserved, so $I_1 = I_2$ and $I_3 = I_4$.
Long wire means more resistance, so $I_3 \lt I_1$.

$I_1 = I_2 \gt I_3 = I_4$

Long Answer Questions

1. Current density and resistivity

When $100$ V is applied across a 5-gauge (diameter $4.62$ mm) wire that is $10$ m long, the magnitude of the current density is $2.0 \times 10^8$ A/m². What is the resistivity of the wire? What is its resistance?

$\displaystyle J = \sigma E = \frac{1}{\rho} E = \frac{1}{\rho} \cdot \frac{V}{\ell}$

$\displaystyle \rho = \frac{V}{J \ell} = 5.0 \times 10^{-8}$ Ω·m

$\displaystyle R = \frac{\rho \ell}{\pi r^2} = 0.030$ Ω

2. Cost of electrical power

An alternative to CFL bulbs and incandescent bulbs are light-emitting diode (LED) bulbs. A $100$ W incandescent bulb can be replaced by a $16$ W LED bulb. Both produce $1600$ lumens of light. Assuming the cost of electricity is $$0.10$ per kilowatt-hour, how much does it cost to run the bulb for one year if it runs for four hours a day?

Energy $\displaystyle E = P \cdot t = 16 \frac{1 \; \rm{kW}}{1000 \; \rm{W}} \cdot 4 \cdot 365 = 23$ kW·h

At $$0.10$ per kW·h, the cost is $$2.30$.

3. Dangerous voltage

Currents of approximately $0.06$ A can be potentially fatal. Currents in that range can make the heart fibrillate (beat in an uncontrolled manner). The resistance of a dry human body can be approximately $100$ kΩ.

What voltage can cause $0.06$ A through a dry human body?
When a human body is wet, the resistance can fall to $100$ Ω. What voltage can cause harm to a wet body?

\(V=IR=6000\) V
\(V=IR=6\) V

4. Resistance and power

A $20.0$ ohm, $5.00$ watt resistor is placed in series with a power supply.

What is the maximum voltage that can be applied to the resistor without harming the resistor?
What would be the current through the resistor?

\(V=\sqrt{PR}=10.0\) V
\(I=V/R=0.50\) A

5. Battery does work

A battery with an emf of $24.00$ V delivers a constant current of $2.00$ mA to an appliance. How much work does the battery do in three minutes?

$\displaystyle W = Pt = IVt = 8.64$ J

Last modified: Mon March 10 2025, 04:39 PM.