# How to judge whether this equation group really has no solution?

I have got the following equation using the following code after the preprocess. The preprocess is complicated so cannot be displayed here in limited scale, and assuming that the following equations are right and following equations are as this:

a = {a1, a2, a3};
b = {b1, b2, b3};
aAirgap = {a1Airgap, a2Airgap, a3Airgap};
bAirgap = {b1Airgap, b2Airgap, b3Airgap};

equa1 = (Br /. {r -> R2}) == (BrAirgap /. {r -> R2})
equa2 = (H\[Theta] /. {r -> R2}) == (H\[Theta]Airgap /. {r -> R2})
equa3 = ((H\[Theta] /. {r -> R1}) == {0, 0, 0})
equa4 = ((H\[Theta]Airgap /. {r -> R3}) == {0, 0, 0})


Then the concrete equations are as follows, in fact, there are 12 equations with 12 unknown variables:

equa1 = ({{(0.36439934858051215 -
0.9312427797057534 I) b2 + (0.36439934858051215 -
0.9312427797057534 I) b3 +
b1 ((0. + 0. I) + (
0.36439934858051215 - 0.9312427797057534 I)/
R2^0.15915492717640267) - (1/R2)
I ((0.36439934858051215 -
0.9312427797057534 I) a2 + (0.36439934858051215 -
0.9312427797057534 I) a3 +
a1 ((0. +
0. I) + (0.36439934858051215 -
0.9312427797057534 I) R2^0.15915492717640267)) +
R2^2. ((0. + 0. I) + (
25.201522001588366 - 1.335913512777944*^-15 I)/(-R1^2 +
R2^2)), (0. + 0. I) +
R2^2. ((0. + 0. I) + (
25.201522001588366 - 1.335913512777944*^-15 I)/(-R1^2 +
R2^2)), (-0.6401843996644798 +
0.768221279597376 I) b1 - (0.6401843996644798 -
0.768221279597376 I) b3 +
b2 ((0. + 0. I) - (
0.6401843996644798 - 0.768221279597376 I)/
R2^0.15915492717640264) - (1/R2)
I ((-0.6401843996644798 +
0.768221279597376 I) a1 - (0.6401843996644798 -
0.768221279597376 I) a3 +

a2 ((0. +
0. I) - (0.6401843996644798 -
0.768221279597376 I) R2^0.15915492717640264)) +
R2^2. ((0. + 0. I) + (
25.201522001588366 - 1.335913512777944*^-15 I)/(-R1^2 +
R2^2))}, {(0.36439934858051215 -
0.9312427797057534 I) b2 + (0.36439934858051215 -
0.9312427797057534 I) b3 +
b1 ((0. + 0. I) + (
0.36439934858051215 - 0.9312427797057534 I)/
R2^0.15915492717640267) - (1/R2)
I ((0.36439934858051215 -
0.9312427797057534 I) a2 + (0.36439934858051215 -
0.9312427797057534 I) a3 +
a1 ((0. +
0. I) + (0.36439934858051215 -
0.9312427797057534 I) R2^0.15915492717640267)),
0. + 0. I, (-0.6401843996644798 +
0.768221279597376 I) b1 - (0.6401843996644798 -
0.768221279597376 I) b3 +
b2 ((0. + 0. I) - (
0.6401843996644798 - 0.768221279597376 I)/
R2^0.15915492717640264) - (1/R2)
I ((-0.6401843996644798 +
0.768221279597376 I) a1 - (0.6401843996644798 -
0.768221279597376 I) a3 +

a2 ((0. +
0. I) - (0.6401843996644798 -
0.768221279597376 I) R2^0.15915492717640264))}, \
{(0.36439934858051215 -
0.9312427797057534 I) b2 + (0.36439934858051215 -
0.9312427797057534 I) b3 +
b1 ((0. + 0. I) + (
0.36439934858051215 - 0.9312427797057534 I)/
R2^0.15915492717640267) - (1/R2)
I ((0.36439934858051215 -
0.9312427797057534 I) a2 + (0.36439934858051215 -
0.9312427797057534 I) a3 +
a1 ((0. +
0. I) + (0.36439934858051215 -
0.9312427797057534 I) R2^0.15915492717640267)) +
R2^2. ((0. + 0. I) - (
25.201522001588366 + 1.335913512777944*^-15 I)/(-R1^2 +
R2^2)), (0. + 0. I) +
R2^2. ((0. + 0. I) - (
25.201522001588366 + 1.335913512777944*^-15 I)/(-R1^2 +
R2^2)), (-0.6401843996644798 +
0.768221279597376 I) b1 - (0.6401843996644798 -
0.768221279597376 I) b3 +
b2 ((0. + 0. I) - (
0.6401843996644798 - 0.768221279597376 I)/
R2^0.15915492717640264) - (1/R2)
I ((-0.6401843996644798 +
0.768221279597376 I) a1 - (0.6401843996644798 -
0.768221279597376 I) a3 +
a2 ((0. +
0. I) - (0.6401843996644798 -
0.768221279597376 I) R2^0.15915492717640264)) +
R2^2. ((0. + 0. I) - (
25.201522001588366 + 1.335913512777944*^-15 I)/(-R1^2 +
R2^2))}} == {{-b1Airgap - b2Airgap - b3Airgap - (
I (-a1Airgap - a2Airgap - a3Airgap))/R2, 0,
b1Airgap + b3Airgap + b2Airgap/R2 - (
I (a1Airgap + a3Airgap + a2Airgap R2))/R2}, {-b1Airgap -
b2Airgap - b3Airgap - (I (-a1Airgap - a2Airgap - a3Airgap))/R2,
0, b1Airgap + b3Airgap + b2Airgap/R2 - (
I (a1Airgap + a3Airgap + a2Airgap R2))/R2}, {-b1Airgap -
b2Airgap - b3Airgap - (I (-a1Airgap - a2Airgap - a3Airgap))/R2,
0, b1Airgap + b3Airgap + b2Airgap/R2 - (
I (a1Airgap + a3Airgap + a2Airgap R2))/R2}});

equa2 = {b2 ((6.10842*10^-17 - 1.56104*10^-16 I)/
R2^1.15915 - (0.364399 - 0.931243 I)/R2^1.) +
b1 (-((0.364399 - 0.931243 I)/R2^1.15915) + (6.10842*10^-17 -
1.56104*10^-16 I)/R2^1.) -
a1 ((6.10842*10^-17 - 1.56104*10^-16 I)/
R2^1. - (0.364399 - 0.931243 I)/R2^0.840845) -
a2 (-((0.364399 - 0.931243 I)/R2^1.) + (6.10842*10^-17 -
1.56104*10^-16 I)/
R2^0.840845) + ((0.364399 - 0.931243 I) a3)/
R2^1. - ((0.364399 - 0.931243 I) b3)/R2^1. -
2 R2 . {(-9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 -
1.82695*10^-47 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 -
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (-5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 +
9.74558*10^-48 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (6.28319 -
2.06542*10^-48 I) ((0. +

0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2))},
b1 ((1.30923*10^-16 - 4.36411*10^-17 I)/
R2^1.15915 + (1.87122*10^-17 - 1.35743*10^-16 I)/R2^1.) +
b2 (-((2.48862*10^-17 + 1.35743*10^-16 I)/
R2^1.15915) + (1.74522*10^-16 - 4.36411*10^-17 I)/R2^1.) -
a2 ((1.74522*10^-16 - 4.36411*10^-17 I)/
R2^1. - (2.48862*10^-17 + 1.35743*10^-16 I)/R2^0.840845) -
a1 ((1.87122*10^-17 - 1.35743*10^-16 I)/
R2^1. + (1.30923*10^-16 - 4.36411*10^-17 I)/
R2^0.840845) - ((1.49635*10^-16 - 1.79384*10^-16 I) a3)/
R2^1. + ((1.49635*10^-16 - 1.79384*10^-16 I) b3)/R2^1. -
2 R2 . {(-9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 -
1.82695*10^-47 I) ((0. +

0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 -
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (-5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 +
9.74558*10^-48 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (6.28319 -
2.06542*10^-48 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2))},
b1 ((1.07314*10^-16 - 1.28777*10^-16 I)/
R2^1.15915 - (0.640184 - 0.768221 I)/R2^1.) +
b2 (-((0.640184 - 0.768221 I)/R2^1.15915) + (1.07314*10^-16 -
1.28777*10^-16 I)/R2^1.) -
a2 ((1.07314*10^-16 - 1.28777*10^-16 I)/
R2^1. - (0.640184 - 0.768221 I)/R2^0.840845) -
a1 (-((0.640184 - 0.768221 I)/R2^1.) + (1.07314*10^-16 -
1.28777*10^-16 I)/
R2^0.840845) + ((0.640184 - 0.768221 I) a3)/
R2^1. - ((0.640184 - 0.768221 I) b3)/R2^1. -
2 R2 . {(-9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 -
1.82695*10^-47 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 -
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (-5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 +
9.74558*10^-48 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (6.28319 -
2.06542*10^-48 I) ((0. +

0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2))}} == {-2 R2 . {0, 0,
0}, -((2500000 a1Airgap)/\[Pi]) + (2500000 b1Airgap)/(\[Pi] \
R2^2) - (2500000 a2Airgap)/(\[Pi] R2) - (2500000 a3Airgap)/(\[Pi] R2) \
+ (2500000 b2Airgap)/(\[Pi] R2) + (2500000 b3Airgap)/(\[Pi] R2) -
2 R2 . {0, 0,
0}, -((2500000 a2Airgap)/\[Pi]) + (2500000 b2Airgap)/(\[Pi] \
R2^2) - (2500000 a1Airgap)/(\[Pi] R2) - (2500000 a3Airgap)/(\[Pi] R2) \
+ (2500000 b1Airgap)/(\[Pi] R2) + (2500000 b3Airgap)/(\[Pi] R2) -
2 R2 . {0, 0, 0}}

equa3 = {b2 ((6.10842*10^-17 - 1.56104*10^-16 I)/
R1^1.15915 - (0.364399 - 0.931243 I)/R1^1.) +
b1 (-((0.364399 - 0.931243 I)/R1^1.15915) + (6.10842*10^-17 -
1.56104*10^-16 I)/R1^1.) -
a1 ((6.10842*10^-17 - 1.56104*10^-16 I)/
R1^1. - (0.364399 - 0.931243 I)/R1^0.840845) -
a2 (-((0.364399 - 0.931243 I)/R1^1.) + (6.10842*10^-17 -
1.56104*10^-16 I)/
R1^0.840845) + ((0.364399 - 0.931243 I) a3)/
R1^1. - ((0.364399 - 0.931243 I) b3)/R1^1. -
2 R1 . {(-9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 -
1.82695*10^-47 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 -
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (-5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 +
9.74558*10^-48 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (6.28319 -
2.06542*10^-48 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2))},
b1 ((1.30923*10^-16 - 4.36411*10^-17 I)/
R1^1.15915 + (1.87122*10^-17 - 1.35743*10^-16 I)/R1^1.) +
b2 (-((2.48862*10^-17 + 1.35743*10^-16 I)/
R1^1.15915) + (1.74522*10^-16 - 4.36411*10^-17 I)/R1^1.) -
a2 ((1.74522*10^-16 - 4.36411*10^-17 I)/
R1^1. - (2.48862*10^-17 + 1.35743*10^-16 I)/R1^0.840845) -
a1 ((1.87122*10^-17 - 1.35743*10^-16 I)/
R1^1. + (1.30923*10^-16 - 4.36411*10^-17 I)/
R1^0.840845) - ((1.49635*10^-16 - 1.79384*10^-16 I) a3)/
R1^1. + ((1.49635*10^-16 - 1.79384*10^-16 I) b3)/R1^1. -
2 R1 . {(-9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 -
1.82695*10^-47 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 -
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (-5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 +
9.74558*10^-48 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (6.28319 -
2.06542*10^-48 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2))},
b1 ((1.07314*10^-16 - 1.28777*10^-16 I)/
R1^1.15915 - (0.640184 - 0.768221 I)/R1^1.) +
b2 (-((0.640184 - 0.768221 I)/R1^1.15915) + (1.07314*10^-16 -
1.28777*10^-16 I)/R1^1.) -
a2 ((1.07314*10^-16 - 1.28777*10^-16 I)/
R1^1. - (0.640184 - 0.768221 I)/R1^0.840845) -
a1 (-((0.640184 - 0.768221 I)/R1^1.) + (1.07314*10^-16 -
1.28777*10^-16 I)/
R1^0.840845) + ((0.640184 - 0.768221 I) a3)/
R1^1. - ((0.640184 - 0.768221 I) b3)/R1^1. -
2 R1 . {(-9.99201*10^-16 +
3.33067*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 -
1.82695*10^-47 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 -
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (-5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) + (6.28319 +
9.74558*10^-48 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2)), (6.28319 -
2.06542*10^-48 I) ((0. +
0. I) - (25.2015 + 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (9.99201*10^-16 +

3.33067*10^-16 I) ((0. +
0. I) - (25.2015 - 1.33591*10^-15 I)/(-R1^2 +
R2^2)) - (5.55112*10^-16 +
6.66134*10^-16 I) ((0. +
0. I) - (4.45305*10^-15 - 7.88861*10^-31 I)/(-R1^2 +
R2^2))}} == {0, 0, 0}

equa4 = {-2 R3 . {0, 0,
0}, -((2500000 a1Airgap)/\[Pi]) + (2500000 b1Airgap)/(\[Pi] \
R3^2) - (2500000 a2Airgap)/(\[Pi] R3) - (2500000 a3Airgap)/(\[Pi] R3) \
+ (2500000 b2Airgap)/(\[Pi] R3) + (2500000 b3Airgap)/(\[Pi] R3) -
2 R3 . {0, 0,
0}, -((2500000 a2Airgap)/\[Pi]) + (2500000 b2Airgap)/(\[Pi] \
R3^2) - (2500000 a1Airgap)/(\[Pi] R3) - (2500000 a3Airgap)/(\[Pi] R3) \
+ (2500000 b1Airgap)/(\[Pi] R3) + (2500000 b3Airgap)/(\[Pi] R3) -
2 R3 . {0, 0, 0}} == {0, 0, 0}


and the parameters are

R1 = N[25/1000];
R2 = N[39/1000];
R3 = N[40/1000];


Finally I use

Solve[{equa1, equa2, equa3, equa4}, {a1, a2, a3, b1, b2, b3, a1Airgap,
a2Airgap, a3Airgap, b1Airgap, b2Airgap, b3Airgap}]


But I cannot get the solution and the result is empty:

{}


So how to judge whether this equation group really doesn't have the solution? And what is the reason?

• You have several typos: R3 . {0, 0, 0} should be R3*{0, 0, 0}? But that is just {0, 0, 0}. And there are a few other "dot products" that don't make sense. You have 3 of the 4 equations named incorrectly: euqa1, euqa2, and eqau4. I'm also not seeing why you have several instances of (0. + 0. I). Rather than Solve you should consider Reduce.
– JimB
May 26, 2023 at 4:39
• @JimB,Thank you, I have corrected them, and as R3 is one constant, so R3 . {0, 0, 0} is equal to R3*{0, 0, 0}, and (0. + 0. I) is generated by the preprocess via Mathematica, so it should be right. And I have used Reduce, and the answer is False, so I do not know what is the reason.
– fhrl
May 26, 2023 at 9:49
• You need to take a closer look at your code. For instance in equa4 I see multiple instances of -2 R3 . {0, 0, 0}. And the dimensions for all 4 equations are wrong. When you combine them together (using Join rather than {equa1, equal2, equa3, equa4}) you should get 12 equations.
– JimB
May 26, 2023 at 16:50