To solve the equation

$1.256 = -\mu_0 N^2 A_g I^2 \left(A_g A_c l_c + 2x\right)^2$

by substituting the given values, let's go through it step by step. Here are the provided parameters:

• $N = 20$
• $\mu_r = 1740$
• $\mu_0 = 1.26 \times 10^{-6} \, \text{Tm/A}$
• $l_c = 0.13885 \, \text{m}$
• $x = \text{air gap} = 0.0004 \, \text{m}$
• $A_g = A_c = 17.5 \times 25 = 430 \, \text{mm}^2 = 0.00043 \, \text{m}^2$

Let's substitute these values into the equation and simplify:

Step 1: Substituting values into the equation

Given:

$1.256 = - (1.26 \times 10^{-6}) \cdot (20)^2 \cdot (0.00043) \cdot I^2 \left(0.00043 \cdot 0.00043 \cdot 0.13885 + 2 \cdot 0.0004\right)^2$

Step 2: Simplifying the expression inside the parentheses

1. Calculate $N^2$: $N^2 = 20^2 = 400$

2. Calculate $A_g \cdot A_c \cdot l_c$: $A_g \cdot A_c \cdot l_c = 0.00043 \cdot 0.00043 \cdot 0.13885$

3. Add $2x$: $2x = 2 \cdot 0.0004 = 0.0008$

Then sum $A_g \cdot A_c \cdot l_c + 2x$ and square this term.

Step 3: Solve for $I^2$