Given that the errors in the measurements are as follows: Length of bar magnet \( =0.1 \mathrm{~cm} \) Breath of bar magnet \( =0.1 \mathrm{~cm} \) Mass of bar magnet \( =0.1 \mathrm{~g} \) Time \( =0.01 \) seconds Calculate the errors in \( M \) and Bo Mass of bar magnet \( =78.7 \mathrm{~g} \) Breath of bar magnet \( =1.5 \mathrm{~cm} \) Time \( =5.062 \) seconds Length of bar magnet \( =7.5 \mathrm{~cm} \) Mes

To calculate the errors in \( M \) (mass) and \( B_0 \) (magnetic field strength), we need to understand how the errors propagate through the calculations. ### Given Data: - Length of bar magnet \( L = 7.5 \, \text{cm} \) with an error \( \Delta L = 0.1 \, \text{cm} \) - Breadth of bar magnet \( B = 1.5 \, \text{cm} \) with an error \( \Delta B = 0.1 \, \text{cm} \) - Mass of bar magnet \( M = 78.7 \, \text{g} \) with an error \( \Delta M = 0.1 \, \text{g} \) - Time \( t = 5.062 \, \text{s} \) with an error \( \Delta t = 0.01 \, \text{s} \) ### Error in Mass \( M \): The error in mass \( M \) is given directly as \( \Delta M = 0.1 \, \text{g} \). ### Error in Magnetic Field Strength \( B_0 \): To calculate the error in \( B_0 \), we need to know the formula used to calculate \( B_0 \). A common formula for the magnetic field strength of a bar magnet is: \[ B_0 = \frac{\mu_0}{4\pi} \cdot \frac{2M}{L^3} \] Where: - \( \mu_0 \) is the permeability of free space (a constant). - \( M \) is the mass of the magnet. - \( L \) is the length of the magnet. To find the error in \( B_0 \), we can use the formula for the propagation of uncertainty. The relative error in \( B_0 \) can be calculated as follows: \[ \frac{\Delta B_0}{B_0} = \sqrt{\left(\frac{\Delta M}{M}\right)^2 + \left(3 \frac{\Delta L}{L}\right)^2} \] ### Step 1: Calculate \( B_0 \) First, we need to calculate \( B_0 \): \[ B_0 = \frac{\mu_0}{4\pi} \cdot \frac{2M}{L^3} \] Assuming \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \): \[ B_0 = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times 78.7 \, \text{g}}{(7.5 \, \text{cm})^3} \] Convert \( M \) to kg and \( L \) to meters: \[ M = 78.7 \, \text{g} = 0.0787 \, \text{kg} \] \[ L = 7.5 \, \text{cm} = 0.075 \, \text{m} \] Now calculate \( B_0 \): \[ B_0 = 10^{-7} \cdot \frac{2 \times 0.0787}{(0.075)^3} \] Calculating \( (0.075)^3 \): \[ (0.075)^3 = 0.000421875 \, \text{m}^3 \] Now substituting back: \[ B_0 = 10^{-7} \cdot \frac{0.1574}{0.000421875} \approx 10^{-7} \cdot 373.76 \approx 3.7376 \times 10^{-5} \, \text{T} \] ### Step 2: Calculate the Errors Now we can calculate the relative errors: 1. **Relative error in mass \( M \)**: \[ \frac{\Delta M}{M} = \frac{0.1}{78.7} \approx 0.00127 \] 2. **Relative error in length \( L \)**: \[ \frac{\Delta L}{L} = \frac{0.1}{7.5} \approx 0.01333 \] 3. **Relative error in \( B_0 \)**: \[ \frac{\Delta B_0}{B_0} = \sqrt{(0.00127)^2 + (3 \cdot 0.01333)^2} \] Calculating \( 3 \cdot 0