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Step-by-Step Solution
Step 1: Recall the formula for maximum torque on an electric dipole
The maximum torque experienced by an electric dipole of dipole moment P in a uniform electric field E is given by:
\tau_{\text{max}} = P \, E \,\sin 90^\circ = P \, E.
Step 2: Identify the given values
We have two electric dipoles with dipole moments:
P_1 = 1.2 \times 10^{-30} \,\text{C}\cdot\text{m}
P_2 = 2.4 \times 10^{-30} \,\text{C}\cdot\text{m}
They are placed in two different uniform electric fields of strengths:
E_1 = 5 \times 10^{4} \,\text{N}\,\text{C}^{-1}
E_2 = 15 \times 10^{4} \,\text{N}\,\text{C}^{-1}
Step 3: Write the ratio of the maximum torques
The ratio of the maximum torques \tau_1 and \tau_2 for the two dipoles is given by:
\displaystyle \frac{\tau_1}{\tau_2} = \frac{P_1 \, E_1}{P_2 \, E_2}.
Step 4: Substitute the numerical values
\displaystyle
\frac{\tau_1}{\tau_2}
= \frac{\bigl(1.2 \times 10^{-30}\bigr)\,\bigl(5 \times 10^{4}\bigr)}{\bigl(2.4 \times 10^{-30}\bigr)\,\bigl(15 \times 10^{4}\bigr)}.
Step 5: Simplify the expression
Combine and simplify the factors:
\displaystyle
\frac{(1.2 \times 5)\,\bigl(10^{-30} \times 10^{4}\bigr)}{(2.4 \times 15)\,\bigl(10^{-30} \times 10^{4}\bigr)}
= \frac{1.2 \times 5}{2.4 \times 15}.
Numerically, this becomes:
\displaystyle \frac{6}{36} = \frac{1}{6}.
Therefore,
\displaystyle \frac{\tau_1}{\tau_2} = \frac{1}{6}.
Step 6: Conclude the value of x
We have \displaystyle \frac{\tau_1}{\tau_2} = \frac{1}{x} = \frac{1}{6}, so the value of x is:
\displaystyle x = 6.
