The critical electric field required to produce electron-positron pairs depends on the physical constants h , c , m_e and e . Use dimensional analysis and assume that the dimensionless coefficient is of order one. The magnitude of the critical electric field, in SI units, is of the order

Step-by-step solution

Let the critical electric field be expressed as E = h^ c^ m_e^ e^ . The dimensional formulas for the given quantities are: [E] = M L T⁻³ A⁻¹ [h] = M L^2 T⁻¹ [c] = L T⁻¹ [m_e] = M [e] = A T Substituting these dimensions into the assumed relation: M L T⁻³ A⁻¹ = (M L^2 T⁻¹)^ (L T⁻¹)^ M^ (A T)^ M L T⁻³ A⁻¹ = M^ + L^ 2 + T^ - - + A^ Equating the powers of the fundamental dimensions on both sides: For A : = -1 For M : + = 1 For L : 2 + = 1 For T : - - + = -3 - - - 1 = -3 + = 2 Solving the system of equations: Subtracting