1. **Problem statement:** A thin conducting plate 3.9 m on a side has a total charge of $-24.0 \times 10^{-6}$ C. We need to find the electric field and force on an electron at 1.0 cm and 3.0 cm above the plate, and the work done when the electron moves between these points. 2. **Formula and concepts:** - The electric field near a large charged plate is approximately uniform and given by: $$E = \frac{\sigma}{2\epsilon_0}$$ where $\sigma = \frac{Q}{A}$ is the surface charge density, $Q$ is total charge, $A$ is area, and $\epsilon_0 = 8.85 \times 10^{-12}$ C$^2$/N·m$^2$ is permittivity of free space. - Force on electron: $F = qE$, where $q = -1.6 \times 10^{-19}$ C. - Work done by electric field moving charge $q$ over distance $d$ in field $E$: $$W = qEd$$ 3. **Calculate area $A$:** $$A = 3.9 \times 3.9 = 15.21 \text{ m}^2$$ 4. **Calculate surface charge density $\sigma$:** $$\sigma = \frac{-24.0 \times 10^{-6}}{15.21} = -1.577 \times 10^{-6} \text{ C/m}^2$$ 5. **Calculate electric field $E$ near the plate:** $$E = \frac{\sigma}{2\epsilon_0} = \frac{-1.577 \times 10^{-6}}{2 \times 8.85 \times 10^{-12}} = -8.91 \times 10^{4} \text{ N/C}$$ Since the plate is large, $E$ is approximately constant at 1.0 cm and 3.0 cm above the plate. 6. **(a) Electric field 1.0 cm above plate:** $$E = -8.91 \times 10^{4} \text{ N/C}$$ (Negative sign indicates direction is downward.) 7. **(b) Force on electron at 1.0 cm:** $$F = qE = (-1.6 \times 10^{-19})(-8.91 \times 10^{4}) = 1.43 \times 10^{-14} \text{ N}$$ (Positive force means upward direction.) 8. **(c) Electric field and force at 3.0 cm:** Electric field remains approximately the same: $$E = -8.91 \times 10^{4} \text{ N/C}$$ Force: $$F = 1.43 \times 10^{-14} \text{ N}$$ 9. **(d) Work done moving electron from 1.0 cm to 3.0 cm:** Since $E$ is uniform, work done by field: $$W = qE \Delta d$$ Distance moved upward: $$\Delta d = 3.0 \text{ cm} - 1.0 \text{ cm} = 0.02 \text{ m}$$ Calculate work: $$W = (-1.6 \times 10^{-19})(-8.91 \times 10^{4})(0.02) = 2.85 \times 10^{-16} \text{ J}$$ **Final answers:** - (a) $E = -8.91 \times 10^{4}$ N/C (downward) - (b) $F = 1.43 \times 10^{-14}$ N (upward) - (c) $E = -8.91 \times 10^{4}$ N/C, $F = 1.43 \times 10^{-14}$ N - (d) $W = 2.85 \times 10^{-16}$ J