9.3.6. а. Through a rectangular plate $a \times b$$(a \ll b)$ passed the current $I$. Modulus of longitudinal elasticity of the plate $E$. Determine by how much the size of $a$ will decrease under the action of magnetic forces.
b. Soft copper "flows" under pressure $4 \cdot 10^7 \, Pa$, and steel - under pressure $5 \cdot 10^8 \, Pa.$ Estimate the minimum magnetic field induction that will cause copper and steel to “flow”.
Solution
а)Energy density (magnetic pressure) is calculated using the formula $$P=w=\frac{B^2}{2\mu_0}$$ Considering the induction to be constant throughout the entire cross-section and equal to $B=\frac{\mu_0I}{2b}$, we get a formula for pressure:$$P=\frac{\mu_0I^2}{8b^2}$$ Hooke's law: $$\frac{\Delta}{a}=\frac{P}{E}$$ $$\Delta=\frac{\mu_0aI^2}{8Eb^2}$$ b)$$P=\frac{B^2}{2\mu_0}$$ $$B=\sqrt{2\mu_0P}$$ $$B_1\approx10Т$$ $$B_2\approx35Т$$