Electric Current

In a gas discharge tube, when the voltage between the two electrodes is sufficiently high, a large number of gas molecules are ionized into electrons and positive ions, making the gas conductive. In a hydrogen gas discharge tube, $3.1 \times 10^{18}$ electrons and $1.1 \times 10^{18}$ hydrogen ions pass through a certain cross-section of the tube per second.

On a dielectric ring of radius $R$, each unit length carries an electric charge $\lambda$. The ring rotates with angular velocity $\omega$ in its own plane about its center, forming a circular current.

In an accelerator, a proton beam of diameter $2.0$ mm forms a current of $1.0$ mA. The proton beam bombards a metal target to produce nuclear reactions. In this proton beam, each proton has kinetic energy $2 \times 10^{7}$ eV. The mass of a proton is $1.7 \times 10^{-27}$ kg.

An electric stove with power $2$ kW is used to heat $2$ kg of water from $20\,^\circ\text{C}$ to $100\,^\circ\text{C}$. The efficiency of the stove is $30\%$. The specific heat of water is $4.2 \times 10^{3}$ J/(kg$\cdot{}^\circ$C).

Two copper wires with different diameters are connected in series and carry a constant current. The ratio of their diameters is $d_1 : d_2 = 3 : 2$, and the ratio of their lengths is $l_1 : l_2 = 1 : 2$.

A linear accelerator produces an electron beam whose current is not steady but consists of pulses. Each pulse lasts $0.1$ $\mu$s with an average current (during the pulse) of $1.6$ A.

The molar mass of copper is $\mu = 63.6$ g/mol and its valence is $n = 2$. A mass of $m = 191$ g of copper is to be deposited from a copper sulfate ($\text{CuSO}_4$) solution by electrolysis. Take Faraday's constant $F = 9.65 \times 10^4$ C/mol, the elementary charge $e = 1.6 \times 10^{-19}$ C, and Avogadro's number $N_A = 6.02 \times 10^{23}$ /mol.

A copper wire with cross-sectional area $S = 2.0\ \text{mm}^2$ carries a current $I = 30$ mA. Assume each copper atom contributes one free electron, and the random thermal speed of the electrons is $v_{\text{th}} = 10^5$ m/s. Useful constants: density of copper $\rho_{\text{Cu}} = 8.9 \times 10^3\ \text{kg/m}^3$, molar mass $\mu = 63.5$ g/mol, resistivity $\rho_e = 1.7 \times 10^{-8}\ \Omega\!\cdot\!\text{m}$, Avogadro's number $N_A = 6.02 \times 10^{23}$ /mol, electron charge $e = 1.6 \times 10^{-19}$ C, electron mass $m_e = 9.1 \times 10^{-31}$ kg.

A copper wire and an iron wire each have length $l = 10$ m and diameter $d = 2.0$ mm. They are connected in series, and a voltage $U = 100$ V is applied across the two ends. Take the resistivities to be $\rho_{\text{Cu}} = 1.7 \times 10^{-8}\ \Omega\!\cdot\!\text{m}$ and $\rho_{\text{Fe}} = 1.0 \times 10^{-7}\ \Omega\!\cdot\!\text{m}$.

When a motor is not running, the resistance of its copper winding at $t_0 = 20\,^{\circ}\!\text{C}$ is $R_0 = 50\ \Omega$. After running for several hours, the resistance rises to $R = 58\ \Omega$. Take the temperature coefficient of resistance for copper to be $\alpha = 4.3 \times 10^{-3}\ {}^{\circ}\!\text{C}^{-1}$.